Introductory Algebra

🧭 Overview

🧠 One-sentence thesis

Real numbers form a hierarchy where rational numbers (including integers, whole numbers, and counting numbers) can be expressed as fractions, ordered, compared, and manipulated using properties like absolute value and opposites.

📌 Key points (3–5)

• Real number hierarchy: Real numbers split into rational (can be written as a/b) and irrational (non-repeating, non-terminating decimals); integers, whole numbers, and counting numbers are all subsets of rational numbers.
• Fractions and equivalence: Proper fractions (numerator < denominator) represent values less than one; improper fractions (numerator ≥ denominator) represent values ≥ one; equivalent fractions have the same value when simplified.
• Ordering and comparing: Rational numbers can be ordered by finding common denominators; integers can be compared on a number line where rightmost is greatest.
• Common confusion: Absolute value always yields a positive result or zero—it measures distance from zero, not direction; don't confuse |−7| with −7.
• Opposites and additive inverse: Every number has an opposite at the same distance from zero in the other direction; a number plus its opposite always equals zero.

🔢 The real number hierarchy

🌳 How real numbers are organized

The excerpt presents real numbers as the most generic category, which then branches into two major types:

Category	Definition	Examples
Rational numbers	Numbers that can be written as a/b where a and b are integers and b ≠ 0	Fractions, integers, terminating decimals
Irrational numbers	Non-repeating, non-terminating decimals	π, √2

🪜 Subsets within rational numbers

Each category is a subset of the one above it:

Counting Numbers: the natural numbers from 1 to infinity, i.e. {1, 2, 3, 4, 5...}

Whole Numbers: all Counting Numbers and the number zero, i.e. {0, 1, 2, 3, 4, 5...}

Integers: all the whole numbers, zero and the negatives of the whole numbers, i.e. {...-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5...}

• All counting numbers are whole numbers.
• All whole numbers are integers.
• All integers are rational numbers.
• All rational and irrational numbers are real numbers.

🍰 Thinking about rational numbers as fractions

The excerpt uses a cake analogy: if you cut a cake into b slices, your share is a of those slices.

Example: The rational number 1/2 means cutting the cake into two parts and taking one part.

🧩 Types and properties of fractions

📏 Proper vs improper fractions

• Proper fractions: numerator is less than the denominator; represents a number less than one; "you always end up with less than a whole cake."
• Improper fractions: numerator is greater than or equal to the denominator; represents a number ≥ one; can be rewritten as a mixed number (an integer plus a proper fraction).

♻️ Equivalent fractions and simplification

Equivalent fractions: two fractions that give the same numerical value.

The excerpt explains that 1/2 and 2/4 are equivalent because they represent the same visual portion.

How to simplify (reduce) a fraction:

1. Write out the prime factors of both numerator and denominator.
2. Cancel matching factors that appear in both.
3. Re-multiply the remaining factors.

Example from the excerpt: (2/4) = (2·1)/(2·2·1) = 1/2

• Reducing does not change the value; it simplifies how we write it.
• When all common factors are canceled, the fraction is in simplest form.

📝 Classification examples

The excerpt provides three examples:

• 3/7: both are prime, so already in simplest form (proper fraction).
• 9/3: improper fraction; simplifies to 3.
• 50/60: proper fraction; simplifies to 5/6 by canceling common factors.

📊 Ordering and comparing rational numbers

🔢 Ordering rational numbers

Ordering rational numbers: arranging them according to directions such as ascending (lowest to highest) or descending (highest to lowest).

The excerpt notes this is useful for determining which unit cost is cheapest.

Example scenario: Three can sizes with different prices—find cost per ounce, then arrange in ascending order: 0.040, 0.061875, 0.07375.

⚖️ Comparing fractions

To compare fractions like 3/7 and 4/9:

1. Create a common denominator (7 × 9 = 63).
2. Convert both fractions: 3/7 = 27/63 and 4/9 = 28/63.
3. Compare numerators: because 28 > 27, therefore 4/9 > 3/7.

📍 Graphing and comparing integers

• To graph an integer on a number line, place a dot above the number.
• Greatest number is farthest to the right; least is farthest to the left.
• Use symbols: > means "greater than"; < means "less than".

Example: 2 is farther right than -5, so 2 > -5 (or -5 < 2).

🎯 Absolute value and opposites

🔄 Opposites and the additive inverse

Opposite: represents the same distance from zero but in the other direction.

Additive Inverse Property: For any real number a, a + (−a) = 0.

• Every number has an opposite.
• Adding a number to its opposite always gives zero.

Example: 7 and -7 are opposites; 7 + (−7) = 0.

📏 Absolute value definition

Absolute value: represents the distance from zero when graphed on a number line.

• Written as |x|, read as "the absolute value of x."
• Both 7 and -7 are 7 units away from zero, so |7| = 7 and |−7| = 7.

⚠️ Key rules for absolute value

• Treat absolute value expressions like parentheses: evaluate operations inside first.
• The absolute value is always positive or zero—it cannot be negative.
• We only care about distance from zero, not direction.

Don't confuse: |−7| = 7 (positive), but −|−7| = −7 (negative sign outside the absolute value).

🧮 Worked examples

The excerpt provides four examples:

• |5 + 4| = |9| = 9
• 3 − |4 − 9| = 3 − |−5| = 3 − 5 = −2
• |−5 − 11| = |−16| = 16
• −|7 − 22| = −|−15| = −(15) = −15

Notice in the last example: the negative sign outside the absolute value makes the final answer negative.

🧭 Overview

🧠 One-sentence thesis

Addition and subtraction of rational numbers follow algebraic properties that allow rearranging and regrouping, and subtraction can always be rewritten as adding the opposite.

📌 Key points (3–5)

• Visual representation: Number lines show addition as jumps to the right (positive) or left (negative).
• Core algebraic properties: Commutative, Associative, and Identity properties allow rearranging, regrouping, and simplifying addition problems.
• Adding fractions: Denominators must be equivalent before adding; mixed numbers should be converted to improper fractions first.
• Subtraction rule: To subtract a number, add its opposite; subtracting a negative is the same as adding a positive (Opposite-Opposite Property).
• Common confusion: Don't confuse the direction of jumps on a number line—positive values jump right, negative values jump left; subtraction always jumps left.

🎯 Addition fundamentals

🏈 Real-world meaning

• Addition combines gains and losses to find a net result.
• Example: A football team gains 11 yards, loses 5 yards, then loses 2 yards → 11 + (−5) + (−2) = 4 yards net gain.
• Losses are negative integers; gains are positive integers.

📏 Number line visualization

• Start at the first number, then "jump" according to the second number.
• Positive addition: jump to the right.
  • Example: −2 + 3 → start at −2, move 3 units right → end at 1.
• Subtraction (negative addition): jump to the left.
  • Example: 2 − 3 → start at 2, move 3 units left → end at −1.
• Don't confuse: the operation determines direction, not just the sign of the number being added.

🧷 Algebraic properties of addition

🔄 Commutative Property

Commutative Property of Addition: For all real numbers a and b, a + b = b + a.

• "Commute" means to change locations—you can rearrange the order of numbers in an addition problem.
• Example: Nadia builds a castle (2 + 1) feet tall; Peter builds (1 + 2) feet tall → both castles are the same height because 2 + 1 = 1 + 2.

🔗 Associative Property

Associative Property of Addition: For all real numbers a, b, and c, (a + b) + c = a + (b + c).

• "Associate" means to group together—you can regroup numbers in an addition problem.
• Example: 9 + (1 + 22) can be regrouped as (9 + 1) + 22 = 10 + 22 = 32, which is easier to compute.

🆔 Identity Property

Identity Property of Addition: For any real number a, a + 0 = a.

• Zero is the additive identity because adding zero to any number returns the original number.
• Example: 4211 + 0 = 4211.

⚖️ Additive Inverse Property

• The sum of a number and its opposite is zero (mentioned as previously learned).

➕ Adding rational numbers

🔢 Converting mixed numbers

• Before adding fractions, convert mixed numbers to improper fractions.
• Method: Multiply the denominator by the whole number, add the numerator, place over the original denominator.
• Example: 11 and 2/3 → (3 × 11) + 2 = 35 → improper fraction is 35/3.

🧮 Addition Property of Fractions

Addition Property of Fractions: For all real numbers a, b, and c, a/c + b/c = (a + b)/c.

• Key rule: Denominators must be equivalent before adding.
• If denominators differ, create a common denominator first.
• Example: 5/6 + 1/18 → convert 5/6 to 15/18 → 15/18 + 1/18 = 16/18 → reduce to 8/9.

➖ Subtraction of rational numbers

🔄 Subtraction as adding the opposite

Rule: To subtract a number, add its opposite.

• Subtraction can always be rewritten as addition of a negative.
• Example: 3 − 5 = 3 + (−5) = −2.
• Example: 9 − 16 = 9 + (−16) = −7.
• On a number line: 9 − 12 → start at 9, move left 12 units → end at −3.

➕ Opposite-Opposite Property

Opposite-Opposite Property: Since taking the opposite of a number changes its sign, −(−b) = b. Therefore, for any real numbers a and b, a − (−b) = a + b.

• Subtracting a negative is the same as adding a positive (double negative becomes positive).
• Example: −6 − (−13) = −6 + 13 = 7.
• Example with fractions: 5/6 − (−1/18) = 5/6 + 1/18 → create common denominator → 15/18 + 1/18 = 16/18 → reduce to 8/9.

🧮 Subtraction steps for fractions

1. Apply the Opposite-Opposite Property if subtracting a negative.
2. Create a common denominator.
3. Add the fractions.
4. Reduce to simplest form.

📊 Practice contexts

Situation	Operation	Result
Blue whale dives 160 feet then rises 8 feet	−160 + 8	Net depth
Temperature −8°F rises 25°F	−8 + 25	New temperature 17°F
Stock price $4.83 drops $0.97	4.83 − 0.97	Closing price $3.86

• Real-world problems often involve combining positive and negative changes.
• Identify gains/rises as positive, losses/drops as negative, then apply addition or subtraction rules.

🧭 Overview

🧠 One-sentence thesis

Multiplying and dividing rational numbers follows specific properties and rules—including sign rules and the reciprocal method—that allow us to solve real-world problems involving fractions and mixed numbers.

📌 Key points (3–5)

• Multiplication of fractions: multiply numerators together and denominators together (a/b · c/d = ac/bd).
• Key multiplication properties: the Multiplication Property of -1, Identity Property (×1), Zero Property (×0), and sign rules (same signs → positive; different signs → negative).
• Division by reciprocals: dividing by a fraction means multiplying by its reciprocal (the "right" reciprocal).
• Common confusion: don't confuse the reciprocal (multiplicative inverse) with the opposite (additive inverse)—reciprocals multiply to 1, opposites add to 0.
• Real-world use: these operations solve practical problems like scaling recipes, calculating speed, and applying physics formulas.

🔢 Properties of Multiplication

🔢 Multiplication Property of -1

For any real number a, (−1) × a = −a.

• In plain language: multiplying any number by negative one gives you the opposite of that number.
• Example: (−1) · 9,876 = −9,876.
• This also works with negative inputs: (−1) · (−322) = 322.

🆔 Identity Property of Multiplication

For any real number a, (1) × a = a.

• Multiplying any value by 1 leaves it unchanged.
• Similar to the Additive Identity (adding 0), but for multiplication.

0️⃣ Zero Property of Multiplication

For any real number a, (0) × a = 0.

• Any value multiplied by zero results in zero.

➕➖ Sign rules for multiplication

Rule	Description	Result
Same Sign Rule	Two positive or two negative numbers	Positive product
Different Sign Rule	One positive and one negative number	Negative product

• Example: (−3) · (−4) = 12 (same signs → positive).
• Example: (−3) · 4 = −12 (different signs → negative).

✖️ Multiplying Rational Numbers

✖️ The fraction multiplication rule

For any real numbers a, b, c, d where b ≠ 0 and d ≠ 0: a/b · c/d = ac/bd

• Multiply the numerators together to get the new numerator.
• Multiply the denominators together to get the new denominator.
• Example: 3/7 · 4/5 = (3·4)/(7·5) = 12/35.

🍪 Whole numbers as fractions

• To multiply a whole number by a fraction, rewrite the whole number as a fraction over 1.
• Example: 8 · 1/3 = 8/1 · 1/3 = 8/3 = 2 2/3.
• Real-world scenario: A recipe calls for 8 cups flour but you need only 1/3 of the recipe → 8 · 1/3 = 2 2/3 cups.

🎨 Visual representation

• Draw one rectangle divided horizontally (for the first fraction) and another divided vertically (for the second fraction).
• Overlay them to create smaller regions; the shaded area represents the product.
• Example: 1/3 · 2/5 = 2/15 (2 shaded regions out of 15 total).

🔄 Associative and Commutative Properties

Associative Property of Multiplication: For any real numbers a, b, and c, (a · b) · c = a · (b · c).

Commutative Property of Multiplication: For any real numbers a and b, a(b) = b(a).

• You can regroup or reorder factors without changing the product.

🧮 Real-world multiplication examples

• Chocolate bar sharing: Anne has 1 bar. Bill takes 1/4, leaving 3/4. Cindy takes 1/3 of what's left: 1/3 × 3/4 = 1/4. Anne and Dora split the remaining 1/2 into two pieces: each gets 1/4.
• Truck mileage: Doris' truck gets 10 2/3 miles per gallon. She fills 5 1/2 gallons. Convert to improper fractions: 32/3 · 11/2 = 352/6 = 58 2/3 miles.

➗ Division of Rational Numbers

🔁 Inverse operations and reciprocals

Inverse operations: Operations that "undo" each other (addition ↔ subtraction; multiplication ↔ division).

Inverse Property of Multiplication: For every nonzero number a, there is a multiplicative inverse 1/a such that a · (1/a) = 1.

Reciprocal: The reciprocal of a nonzero rational number a/b is b/a.

• Two nonzero numbers whose product is 1 are reciprocals.
• Important: Zero does not have a reciprocal.
• Don't confuse: the reciprocal is not the same as the opposite. The reciprocal of 3 is 1/3 (they multiply to 1); the opposite of 3 is −3 (they add to 0).

➗ The "right" reciprocal rule

Rule: When dividing rational numbers, multiply by the reciprocal of the fraction on the right-hand side of the division operator.

• Example: 2/9 ÷ 3/7 = 2/9 × 7/3 = 14/27.
• Example: 7/3 ÷ 2/3 = 7/3 × 3/2 = 7/2.

📏 Large fraction bar notation

• A large fraction bar separating two fractions indicates division.
• Example: (2/3) over (7/8) means 2/3 ÷ 7/8 = 2/3 × 8/7 = 16/21.

🧪 Real-world division examples

• Newton's Second Law: a = F/m. If F = 7 1/3 and m = 1/5, convert to improper fractions: a = 22/3 ÷ 1/5 = 22/3 × 5/1 = 110/3 = 36 2/3 m/s².
• Running speed: Anne runs 1.5 miles in 1/4 hour. Speed = distance/time = 1.5 ÷ 1/4 = 3/2 × 4/1 = 6 mi/hr.
• Paint coverage: A can covers 50 sq ft per pint. A 1/8 pint sample covers 2 ft × 3 ft = 6 sq ft. Expected coverage: 50 × 1/8 = 6.25 sq ft. Actual is less than stated.
• Trench digger: Machine moves at 3/8 mph. To dig 2/3 mile: time = distance/speed = 2/3 ÷ 3/8 = 2/3 × 8/3 = 16/9 hours.

🔍 Common Confusions

🔍 Reciprocal vs opposite

• Reciprocal (multiplicative inverse): a number that multiplies with the original to give 1.
  • Example: reciprocal of 5 is 1/5 because 5 × 1/5 = 1.
• Opposite (additive inverse): a number that adds with the original to give 0.
  • Example: opposite of 5 is −5 because 5 + (−5) = 0.
• These are not the same concept; don't mix them up.

🔍 Why zero has no reciprocal

• A reciprocal of a must satisfy a · (1/a) = 1.
• For zero: 0 · (anything) = 0, never 1.
• Therefore, zero cannot have a reciprocal.

🧭 Overview

🧠 One-sentence thesis

The order of operations is a systematic method for evaluating numerical expressions that ensures everyone arrives at the same correct answer by performing operations in a specific sequence: parentheses, exponents, multiplication/division (left to right), then addition/subtraction (left to right).

📌 Key points (3–5)

• What it is: A fixed sequence (P-E-MD-AS) that tells you which operation to perform first when an expression contains multiple operations.
• Why it matters: Without following the order of operations, the same expression can yield different answers—only one is correct.
• Common confusion: Don't work left to right automatically; multiplication and division come before addition and subtraction, even if they appear later in the expression.
• Grouping symbols change priority: Parentheses override the normal order—always evaluate what's inside parentheses first.
• Checking work: You can verify whether an answer is correct by re-evaluating the expression using the order of operations.

🔢 Expressions vs Equations

🔢 What is an expression?

An expression is a number sentence without an equals sign. It can be simplified and/or evaluated.

• Example from the excerpt: 4 + 3 × 5
• Contrast with an equation, which has an equals sign and can be solved (e.g., 3 + 4 = 7).
• Expressions don't claim two sides are equal; they are single mathematical phrases that produce a value when evaluated.

📋 The Order of Operations Rule

📋 The PEMDAS sequence

The excerpt presents the order as:

Step	Operation	Notes
P	Parentheses	Always first
E	Exponents	Second
MD	Multiplication or Division	Left to right, whichever comes first
AS	Addition or Subtraction	Left to right, whichever comes first

• Key point: Multiplication and division have equal priority—you perform them in the order they appear from left to right, not "all multiplication then all division."
• Same rule applies to addition and subtraction.

⚠️ What happens without the order of operations?

The excerpt demonstrates with 4 + 3 × 5:

• Correct (following order of operations): Multiply first: 3 × 5 = 15, then add: 4 + 15 = 19.
• Incorrect (left to right): Add first: 4 + 3 = 7, then multiply: 7 × 5 = 35.

Don't confuse: The "natural" left-to-right reading order is not the mathematical order of operations.

🧮 Working with Parentheses and Exponents

🧮 Parentheses come first

• Parentheses (grouping symbols) override all other operations.
• Always complete the work inside parentheses before anything else.
• Example from the excerpt: 2 + (3 − 1) × 2
  • First: 3 − 1 = 2 (inside parentheses)
  • Then: 2 × 2 = 4 (multiplication)
  • Finally: 2 + 4 = 6 (addition)

🧮 Exponents come second

• After parentheses, evaluate any exponents.
• Example from the excerpt: 35 + 3² − (3 × 2) × 7
  • Parentheses first: 3 × 2 = 6
  • Exponents second: 3² = 3 × 3 = 9
  • Then multiplication: 6 × 7 = 42
  • Finally addition/subtraction left to right: 35 + 9 = 44, then 44 − 42 = 2

⚠️ Multiple operations in one expression

When an expression has parentheses, exponents, multiplication, and addition/subtraction all together, work through P-E-MD-AS in strict order.

Don't confuse: Even if addition appears before multiplication in the written expression, you still multiply first (unless the addition is inside parentheses).

🔍 Checking Answers (Being a "Math Detective")

🔍 How to verify correctness

The excerpt describes using the order of operations to check someone else's work:

• Re-evaluate the expression yourself following P-E-MD-AS.
• Compare your result to the given answer.
• If they match, the work is correct; if not, identify where the mistake occurred.

🔍 Common errors in the excerpt's examples

The excerpt shows a student named Joaquin making mistakes:

• Problem 3 error: Joaquin got 19 instead of 13 because he worked left to right instead of multiplying first.
• Problem 5 error: Joaquin got 66 instead of 30 because he subtracted before multiplying.

Lesson: Most errors come from ignoring the order of operations and just working left to right.

🎯 Inserting Parentheses to Make Statements True

🎯 How grouping symbols change the answer

• Placing parentheses in different positions changes which operations are performed first.
• This can change the final result.

🎯 Example from the excerpt

Expression: 5 + 3 × 2 + 7 − 1

• Without parentheses: Following order of operations gives 17.
• Goal: Make the answer equal 22.
• Solution: Insert parentheses around 5 + 3: (5 + 3) × 2 + 7 − 1
  • Parentheses first: 5 + 3 = 8
  • Multiply: 8 × 2 = 16
  • Add/subtract left to right: 16 + 7 − 1 = 22 ✓

Strategy: You may need to try parentheses in different positions to achieve the target answer.

🐦 Real-World Application: The Aviary Problem

🐦 The scenario

• Starting count: 256 birds
• Changes: 3 birds each gave birth to 5 babies; 2 birds released; 3 new injured birds found
• Expression: 256 + 3 × 5 − 2 + 3

🐦 Keisha's mistake

Keisha calculated 1,296 birds—incorrect because she didn't follow the order of operations.

🐦 Correct solution

• Multiply first: 3 × 5 = 15
• Then work left to right: 256 + 15 − 2 + 3
• Step by step: 256 + 15 = 271, then 271 − 2 = 269, then 269 + 3 = 272

Answer: 272 birds in the aviary.

Takeaway: Real-world problems require the order of operations to get the correct numerical result.

🧭 Overview

🧠 One-sentence thesis

This review consolidates skills in comparing and ordering real numbers, performing operations with fractions and decimals, applying arithmetic properties, and solving real-world problems using these foundational concepts.

📌 Key points (3–5)

• Comparing and ordering: determine which real numbers are larger and arrange them from least to greatest, including fractions, decimals, and negative numbers.
• Fraction and decimal operations: add, subtract, multiply, and divide fractions (including mixed numbers) and decimals (including negative values).
• Properties of operations: recognize and apply properties like the commutative, associative, and inverse properties.
• Common confusion: mixed operations require careful attention to signs (negative vs. positive) and proper conversion between fractions and decimals.
• Real-world application: translate word problems into arithmetic expressions and solve for unknown quantities.

🔢 Comparing and Ordering Real Numbers

🔢 Determining which number is larger

• Compare two real numbers to identify the largest value.
• Types include whole numbers, fractions, decimals, and negative numbers.
• Example: comparing 7 and -11 requires understanding that positive numbers are always greater than negative numbers.

📊 Ordering from least to greatest

• Arrange multiple real numbers in ascending order.
• May require converting fractions to common denominators or decimal form for easier comparison.
• Example: ordering fractions like 8/11, 7/10, and 5/9 requires finding equivalent forms or decimal approximations.

📍 Graphing on a number line

• Plot values including mixed numbers, negative decimals, fractions, repeating decimals, and negative fractions.
• Visual representation helps clarify relative positions and magnitudes.

➕ Operations with Fractions and Mixed Numbers

➕ Addition and subtraction

• Combine fractions by finding common denominators.
• Handle mixed numbers by converting to improper fractions or working with whole and fractional parts separately.
• Example: 8/5 − 4/3 requires finding a common denominator before subtracting.

✖️ Multiplication and division

• Multiply fractions by multiplying numerators and denominators directly.
• Divide fractions by multiplying by the reciprocal of the divisor.
• Mixed numbers must be converted to improper fractions first.
• Example: −1 5/7 × −2 1/2 involves converting both mixed numbers, then multiplying (note: two negatives make a positive).

⚠️ Working with negative fractions

• Pay attention to signs throughout the operation.
• Subtracting a negative is equivalent to adding a positive.
• Example: 1 1/5 − (−3 3/4) becomes addition: 1 1/5 + 3 3/4.

🔢 Operations with Decimals

🔢 Adding and subtracting decimals

• Align decimal points and combine values.
• Handle negative decimals carefully: adding two negatives yields a more negative result; subtracting a negative is equivalent to addition.
• Example: (−7.1) + (−0.4) = −7.5 (both negative, so sum is more negative).
• Example: 1.58 − (−13.6) = 1.58 + 13.6 = 15.18 (subtracting a negative becomes addition).

🔄 Multiple operations

• Work left to right, handling signs at each step.
• Example: (−13.6) + 12 − (−15.5) requires processing each operation in sequence, converting subtracting negatives to addition.

🧩 Properties of Operations

🧩 Recognizing properties

The excerpt asks students to identify which property has been applied in given equations:

Property type	Example from excerpt	What it shows
Inverse property	6.78 + (−6.78) = 0	A number plus its opposite equals zero
Commutative property	9.8 + 11.2 + 1.2 = 9.8 + 1.2 + 11.2	Order of addition can be rearranged
Subtraction as addition	4/3 − (−5/6) = 4/3 + 5/6	Subtracting a negative equals adding the positive
Commutative (multiplication)	8(11)(1/8) = 8(1/8)(11)	Order of multiplication can be rearranged

🔍 Why properties matter

• Recognizing properties helps simplify calculations and verify correctness.
• Understanding that subtraction of a negative is addition prevents sign errors.

🌍 Real-World Applications

🌍 Unit conversion problems

• Example: Carol has 18 feet of fencing plus 132 inches—requires converting to a common unit (feet or inches) before adding.
• Real-world problems often mix units and require conversion before calculation.

🍪 Division problems with fractions

• Example: Each cookie requires 3/8 pound of dough; with 12 pounds available, how many cookies can be made?
• Solution approach: divide total amount by amount per item (12 ÷ 3/8).

⏱️ Rate and distance problems

• Example: A machine moves at 3/8 miles/hour; how long to travel 14 miles?
• Requires dividing distance by rate to find time.

💰 Working backward from a final amount

• Example: Georgia spent specific amounts and has $0.16 left; find the starting amount.
• Add all expenditures plus the remaining amount to find the original value.

📐 Area and measurement

• Example: A square garden has area 145 square meters; find the side length exactly.
• Uses the formula A = s² and requires finding the square root (exact form, not decimal approximation).

🧭 Overview

🧠 One-sentence thesis

Variable expressions allow us to represent unknown or changing quantities with symbols and evaluate them by substituting specific values, making it possible to model real-life situations mathematically.

📌 Key points (3–5)

• Variables as symbols: Letters (usually English letters) replace unknown or changing quantities in mathematical situations.
• Evaluation process: To evaluate an expression, substitute a number for the variable, then complete the operations.
• Alternative multiplication notation: To avoid confusion between the multiplication symbol × and the variable x, use parentheses, a dot (·), or write expressions side by side.
• Common confusion: "Evaluate," "simplify," and "answer" all mean the same thing—complete the operations in the expression.
• Real-world application: Variables help solve practical problems like calculating earnings, measuring distances, or determining material quantities.

🗣️ Mathematics as a language

🗣️ Operations as action words

• Mathematics is described as a language that must be learned and practiced.
• Operations in math are like verbs in English—they involve "doing something."
• Familiar operations include addition, subtraction, multiplication, and division.
• More complex operations include raising to an exponent or taking a square root.

📝 Multiple words, same meaning

• Just as English uses synonyms, mathematics uses different words for the same operation.
• Example: "sum," "addition," "more than," and "plus" all mean to add numbers together.
• Similarly, "evaluate," "simplify," and "answer" all mean to complete the operations in an expression.

🔤 Understanding variables

🔤 What a variable is

A variable is a symbol, usually an English letter, written to replace an unknown or changing quantity.

• Variables represent values that are not yet known or that can change.
• Choosing meaningful variable symbols helps clarify what they represent.

💡 Choosing appropriate variables

The excerpt provides examples of good variable choices:

Situation	Changing quantity	Good variable choice
Number of cars on a road	Cars	c
Time in minutes of a ball bounce	Time	t
Distance from an object	Distance	d

• The key is to pick a letter that relates to what you're measuring.
• Example: If you earn $8.15 per hour, let h represent hours worked, and write: amount of money = 8.15h.

🔄 Substitution and evaluation

🔄 What substitution means

To substitute means to replace the variable in the sentence with a value.

• Substitution is the first step in evaluating an expression.
• You take the variable symbol and replace it with an actual number.

🧮 What evaluation means

To evaluate means to complete the operations in the math sentence.

• After substituting, you perform the arithmetic operations.
• Example: Evaluate 7y − 11 when y = 4.
  • Substitute: 7 × 4 − 11
  • Multiply: 28 − 11
  • Subtract: 17
  • The solution is 17.

⚠️ Don't confuse evaluation terms

• "Evaluate," "simplify," and "answer" all mean the same thing in this context.
• They all instruct you to complete the operations and find the result.

✖️ Multiplication notation alternatives

✖️ Why alternative symbols matter

• The multiplication symbol × is easily confused with the variable x.
• To avoid confusion, mathematicians use different notation.

🔀 Three alternative notations

The excerpt shows how to rewrite P = 2 × l + 2 × w:

1. Multiplication dot: P = 2 · l + 2 · w
2. Side-by-side writing: P = 2l + 2w
3. Parentheses: (implied in the context, though not explicitly shown in this example)

• All three notations mean exactly the same thing.
• Side-by-side writing (2l) is the most compact and commonly used.

🐴 Real-world application example

🐴 Fencing a rectangular pasture

The excerpt provides a practical scenario:

• A rectangular pasture measures 300 feet by 225 feet.
• Question: How much fencing is needed to enclose it?

📐 Solution process

1. Draw and label: Sketch the rectangle with dimensions.
2. Write the expression: L + L + W + W (add all four sides).
3. Substitute values: 300 + 300 + 225 + 225.
4. Evaluate: Add the values together = 1,050 feet.
5. Answer: The ranch hand must purchase 1,050 feet of fencing.

• This shows how variables (L for length, W for width) represent real measurements.
• Substitution and evaluation turn the abstract expression into a concrete answer.

🧭 Overview

🧠 One-sentence thesis

Algebraic expressions translate real-world patterns and English phrases into mathematical notation by combining numbers, variables, and operations.

📌 Key points (3–5)

• What an algebraic expression is: a mathematical phrase combining numbers and/or variables using operations.
• Translation process: converting English phrases (like "the product of c and 4") into mathematical symbols (like 4c).
• Real-world application: expressions model practical situations, such as calculating revenue based on the number of people.
• Common confusion: distinguishing when to use parentheses—"3 times the sum of c and 4" becomes 3(c + 4), not 3c + 4.
• Multiple notations: multiplication can be written as ×, ·, or by placing terms side by side (2 × l = 2 · l = 2l).

📝 What algebraic expressions are

📝 Definition and purpose

An algebraic expression is a mathematical phrase combining numbers and/or variables using mathematical operations.

• Expressions describe patterns in numbers using the mathematical verbs (operations) and variables from earlier lessons.
• They are not complete equations—they are phrases, not sentences with an equals sign.
• Purpose: to represent quantities and relationships in a compact, mathematical form.

🔤 Multiple ways to write multiplication

The excerpt shows three equivalent notations:

Notation	Example
× symbol	P = 2 × l + 2 × w
· symbol	P = 2 · l + 2 · w
Side by side	P = 2l + 2w

• All three mean the same thing.
• The side-by-side notation (2l) is shorthand and most common in algebra.

🔄 Translating English to algebra

🔄 Basic translation steps

The process involves identifying:

1. The verb (operation): product, sum, quotient, difference, times, divided by, etc.
2. The nouns (numbers and variables): specific numbers or unknown quantities.
3. The order and grouping: what gets combined first.

Example: "The product of c and 4"

• Verb: "product" means multiply
• Nouns: c and 4
• Result: 4 × c, 4(c), or 4c

🎯 Real-world example: theme park revenue

The excerpt presents a theme park charging $28 per person.

• English phrase: "Twenty-eight times the number of people who enter the park"
• Choose a variable: let p = number of people
• Translation: 28 × p or 28p
• This expression models the total revenue pattern.

Don't confuse: The expression 28p is not a final answer—it's a formula that gives different results depending on the value of p.

🧮 Handling compound phrases

When phrases contain multiple operations, parentheses show what to do first.

Example: "3 times the sum of c and 4"

• Break it down: "3 times" followed by "(the sum of c and 4)"
• The sum must be calculated first: (c + 4)
• Then multiply by 3: 3(c + 4)

Common mistake: Writing 3c + 4 instead of 3(c + 4)

• These are different!
• 3(c + 4) means "add c and 4 first, then multiply the result by 3"
• 3c + 4 means "multiply c by 3, then add 4"

🗣️ Common phrase patterns

🗣️ Key mathematical verbs

The excerpt provides several translation examples:

English phrase	Mathematical verb	Algebraic expression
The product of k and three	multiply	3k
The quotient of h and 8	divide	h ÷ 8
The sum of g and -7	add	g + (-7)
r minus 5.8	subtract	r - 5.8
6 more than 5 times a number	multiply, then add	5n + 6
6 divided by a number	divide	6 ÷ n

🔢 Order matters

• "Sixteen more than a number" → n + 16 (not 16 + n, though they're equivalent)
• "Forty-two less than y" → y - 42 (not 42 - y, which would be different!)
• For subtraction and division, the order of the English phrase determines the order in the expression.

🧭 Overview

🧠 One-sentence thesis

Simplifying algebraic expressions by combining like terms makes expressions smaller and simpler without solving them, but only terms with the same variable raised to the same power can be combined.

📌 Key points (3–5)

• What simplifying means: making an expression smaller or simpler without solving it—you're reducing the number of terms, not finding a value.
• What a term is: a number alone, a variable alone, or a number multiplied by a variable (e.g., 4x, 3, or y).
• Like terms definition: terms that use the same variable raised to the same power can be combined; terms with different variables or powers cannot.
• Common confusion: don't confuse simplifying with evaluating—simplifying combines like terms; evaluating requires substituting a value for the variable.
• How to combine: add or subtract only the numerical parts of like terms; the variable part stays the same as a "label."

🔤 Understanding terms and expressions

🔤 What is a term

A term can be either a number or a variable or it can be a number multiplied by a variable.

• In the expression 4x + 3, there are two terms:
  • First term: 4x (number times variable)
  • Second term: 3 (number alone)
• Each part separated by addition or subtraction is a separate term.
• Without a value for the variable, the expression stays as is unless like terms can be combined.

🎯 What makes terms "like"

A like term means that the terms in question use the same variable, raised to the same power.

Examples of like terms:

• 4x and 5x are like terms (both have x)
• 7y and 2y are like terms (both have y)

Examples of unlike terms:

• 6x and 2y are NOT like terms (different variables)
• 3x and 3x² are NOT like terms (different powers)

Key insight: Think of the variable as a label that tells you whether terms are alike—only terms with matching labels can be combined.

➕ Combining like terms with addition

➕ Adding like terms

When expressions have like terms, you can simplify by adding the numerical parts together while keeping the variable the same.

Example: 5x + 7x

• Check: both terms have x, so they are alike
• Add the numbers: 5 + 7 = 12
• Keep the variable: 12x
• The x stays the same throughout

🔀 Mixed terms with addition

Example: 7x + 2x + 5y

• Identify like terms: 7x and 2x are alike (both have x); 5y is different
• Combine only the like terms: 7x + 2x = 9x
• Keep unlike terms separate: 5y cannot be combined
• Final answer: 9x + 5y

Don't confuse: You cannot add terms with different variables—9x + 5y cannot be simplified further because x and y are not alike.

➖ Combining like terms with subtraction

➖ Subtracting like terms

The same principle applies to subtraction: subtract the numerical parts and keep the variable.

Example: 9y - 2y

• Check: both terms have y, so they are alike
• Subtract the numbers: 9 - 2 = 7
• Keep the variable: 7y

➕➖ Mixed operations

Example: 8x - 3x + 2y + 4y

• Group like terms: (8x - 3x) and (2y + 4y)
• Simplify each group:
  • 8x - 3x = 5x
  • 2y + 4y = 6y
• Combine results: 5x + 6y

Remember: You can only combine terms that are alike—this is the fundamental rule for simplifying expressions.

⚠️ Key reminders

⚠️ What simplifying is NOT

What simplifying IS	What simplifying is NOT
Making an expression smaller by combining like terms	Solving for a variable's value
Reducing the number of terms	Evaluating by substituting numbers
Works without knowing variable values	Requires a value for the variable

🎓 Critical rule

You can ONLY combine terms that are alike.

This means:

• Same variable (x with x, y with y)
• Same power (x with x, not x with x²)
• The variable acts as a label showing which terms can be combined
• Unlike terms must remain separate in the final answer

🧭 Overview

🧠 One-sentence thesis

The Distributive Property allows us to multiply a number or expression across terms inside parentheses, and it is essential for simplifying algebraic expressions and solving real-world problems.

📌 Key points (3–5)

• What the property states: A(B + C) = AB + AC and A(B - C) = AB - AC for any real numbers or expressions.
• How to apply it: multiply the outside term by each term inside the parentheses separately, then combine like terms.
• Works with fractions and negatives: the property applies when distributing fractions, negative signs, and variables.
• Common confusion: when subtracting a polynomial, you must distribute -1 to every term inside the parentheses, not just the first term.
• Why it matters: the Distributive Property appears frequently in everyday applications like business calculations and geometry problems.

📐 The core property and basic examples

📐 Definition and formula

The Distributive Property: For any real numbers or expressions A, B and C:
A(B + C) = AB + AC
A(B - C) = AB - AC

• This property lets you "distribute" multiplication over addition or subtraction.
• You multiply the outside term by each term inside the grouping symbols.

🔢 Verifying with numbers

The excerpt shows that both order of operations and the Distributive Property give the same result.

Example: 11(2 + 6)

• Using order of operations: 11(2 + 6) = 11(8) = 88
• Using Distributive Property: 11(2) + 11(6) = 22 + 66 = 88

Both methods produce 88, confirming the property works.

🧮 With algebraic expressions

Example: Simplify 7(3x - 5)

Two ways to think about it:

• As repeated addition: write (3x - 5) seven times and add all like terms → 21x - 35
• Using the property: 7(3x) + 7(-5) = 21x - 35

Both approaches yield the same simplified expression.

🔧 Applying the property in different contexts

🔧 With fractions as coefficients

Example: Simplify (2/7)(3y² - 11)

• Distribute 2/7 to each term: (2/7)(3y²) + (2/7)(-11)
• Result: (6y²)/7 - 22/7

The property works the same way even when the multiplier is a fraction.

📊 With fraction bars as grouping symbols

The fraction bar acts as a grouping symbol, so the Distributive Property applies.

Example: Simplify (2x + 4)/8

• Rewrite as (1/8)(2x + 4)
• Distribute: (1/8)(2x) + (1/8)(4)
• Simplify: 2x/8 + 4/8 = x/4 + 1/2

This shows how to break apart fractions with sums or differences in the numerator.

⚠️ Distributing negative signs

When subtracting a polynomial, you must distribute -1 to every term.

Example: Simplify 3 - (2x - 4)

• Rewrite as 3 + (-1)(2x - 4)
• Distribute the -1: 3 + (-1)(2x) + (-1)(-4)
• Simplify: 3 - 2x + 4 = -2x + 7

Don't confuse: A common error is writing 3 - (2x - 4) as 3 - 2x - 4. You must change the sign of every term inside the parentheses.

🏗️ Real-world applications

🏗️ Construction and geometry problems

The excerpt provides a practical scenario involving building materials.

Example: An octagonal gazebo needs steel supports—five-foot supports for the base and four-foot supports for the roof-line on each of eight sides.

• Steel required = 8(4 + 5) feet
• Using the Distributive Property: 8 × 4 + 8 × 5 = 32 + 40 = 72 feet total

This shows how the property helps calculate totals when you have multiple groups of mixed items.

📚 Inventory problems

Example: A bookshelf has five shelves, each containing seven poetry books and eleven novels.

• Total books = 5(7 + 11)
• Distribute: 5(7) + 5(11) = 35 poetry books + 55 novels

The Distributive Property simplifies counting when items are organized in identical groups.

💡 Mental math shortcuts

The property can make mental calculations easier.

Example: Simplify 6(19.99) in your head

• Rewrite as 6(20 - 0.01)
• Distribute: 6(20) - 6(0.01) = 120 - 0.06 = 119.94

This technique works well when numbers are close to round values.

🔄 Combining with like terms

🔄 Distribute first, then simplify

When an expression has both distribution and like terms, follow the order: distribute first, then combine.

Example: Simplify 2 + 4(6x - 4)

• Distribute the 4: 2 + 4(6x) + 4(-4) = 2 + 24x - 16
• Combine like terms: 24x + (2 - 16) = 24x - 14

Don't confuse: You cannot add the 2 and 4 before distributing, because the 4 is multiplying the parentheses, not being added to 2.

➕ Adding polynomials

When adding polynomials, remove parentheses (no sign change needed) and combine like terms.

Example: Add (3x² - 4x + 7) and (2x³ - 4x² - 6x + 5)

• Write the sum: (3x² - 4x + 7) + (2x³ - 4x² - 6x + 5)
• Group like terms: 2x³ + (3x² - 4x²) + (-4x - 6x) + (7 + 5)
• Simplify: 2x³ - x² - 10x + 12

➖ Subtracting polynomials

When subtracting, distribute -1 to every term in the second polynomial.

Example: Subtract (x³ - 3x² + 8x + 12) from (4x² + 5x - 9)

• Write as: (4x² + 5x - 9) - 1(x³ - 3x² + 8x + 12)
• Distribute -1: 4x² + 5x - 9 - x³ + 3x² - 8x - 12
• Group like terms: -x³ + (4x² + 3x²) + (5x - 8x) + (-9 - 12)
• Simplify: -x³ + 7x² - 3x - 21

Common error: The excerpt mentions a student who rewrote 4(9x + 10) as 36x + 10, forgetting to distribute the 4 to the second term. The correct answer is 36x + 40.

🧭 Overview

🧠 One-sentence thesis

Adding and subtracting polynomials requires distributing signs correctly and combining like terms to simplify the result.

📌 Key points (3–5)

• Addition rule: Write the sum of polynomials and combine like terms by grouping terms with the same variable and exponent.
• Subtraction rule: Distribute a negative one (−1) to every term of the polynomial being subtracted, then combine like terms.
• Common confusion: When subtracting, you must distribute the negative sign to every term in the second polynomial, not just the first term.
• Checking your work: Substitute convenient values (not 0 or 1) for variables into both the original expression and your answer; if they evaluate to the same number, your answer is likely correct.
• Application: Polynomial addition and subtraction can represent areas of geometric figures by adding or subtracting areas of component shapes.

➕ Adding polynomials

➕ The addition process

To add polynomials:

• Write the sum of all polynomials.
• Group terms that have the same variable raised to the same power (like terms).
• Combine the coefficients of like terms.

📝 Example walkthrough

Problem: Add 3x² − 4x + 7 and 2x³ − 4x² − 6x + 5

Steps:

• Write the sum: (3x² − 4x + 7) + (2x³ − 4x² − 6x + 5)
• Group like terms: 2x³ + (3x² − 4x²) + (−4x − 6x) + (7 + 5)
• Simplify: 2x³ − x² − 10x + 12

The key is identifying which terms share the same variable and exponent, then adding their coefficients.

➖ Subtracting polynomials

➖ The subtraction process

To subtract one polynomial from another:

• Distribute −1 to every term of the polynomial you are subtracting.
• This changes the sign of every term in the second polynomial.
• Then group and combine like terms as in addition.

🔄 Why distribute the negative

When subtracting, distribute a −1 to each term of the polynomial you are subtracting.

The excerpt shows this step explicitly:

• Original: (4x² + 5x − 9) − (x³ − 3x² + 8x + 12)
• After distributing −1: 4x² + 5x − 9 − x³ + 3x² − 8x − 12
• Notice every term in the second polynomial flipped its sign.

Don't confuse: Subtracting a polynomial is not the same as just changing the sign of the first term; you must change the sign of all terms.

📝 Example with two variables

Problem: Subtract 5b² − 2a² from 4a² − 8ab − 9b²

Steps:

• Write as: (4a² − 8ab − 9b²) − (5b² − 2a²)
• Distribute −1: 4a² − 8ab − 9b² − 5b² + 2a²
• Group like terms: (4a² + 2a²) + (−9b² − 5b²) − 8ab
• Simplify: 6a² − 14b² − 8ab

✅ Checking your answer

✅ Substitution method

After simplifying, you can verify your work:

• Pick convenient values for each variable (avoid 0 or 1).
• Substitute these values into the original expression.
• Substitute the same values into your simplified answer.
• If both evaluate to the same number, your answer is likely correct.

📝 Example check

For the problem (4a² − 8ab − 9b²) − (5b² − 2a²) = 6a² − 14b² − 8ab, let a = 2 and b = 3:

Expression	Calculation	Result
Original: (4(2)² − 8(2)(3) − 9(3)²) − (5(3)² − 2(2)²)	(16 − 48 − 81) − (45 − 8) = (−113) − 37	−150
Answer: 6(2)² − 14(3)² − 8(2)(3)	24 − 126 − 48	−150

Both evaluate to −150, so the answer is correct.

Important: Do not use 0 or 1 for checking, as these values can hide errors.

📐 Application to geometric areas

📐 Adding areas

When a figure is made of multiple shapes, find the total area by:

• Calculating the area of each component shape.
• Adding all the individual areas together.
• Combining like terms in the resulting polynomial.

📝 Example: composite figure

A figure made of two squares and two rectangles:

• Blue square: y × y = y²
• Yellow square: x × x = x²
• Two pink rectangles: each x × y = xy

Total area = y² + x² + xy + xy = y² + x² + 2xy

📐 Subtracting areas

When finding the area of a region with a hole or cutout:

• Calculate the area of the larger shape.
• Subtract the area of the smaller shape or cutout.

📝 Example: square with cutout

Green region = big square minus little square:

• Big square: y × y = y²
• Little square: x × x = x²
• Green region area: y² − x²

🔄 Multiple methods

Some figures can be calculated in more than one way:

• Method 1: Add all the component areas directly.
• Method 2: Find a larger enclosing shape and subtract unwanted areas.

Example: A figure can be found as 6a² either by adding six small squares (a² + a² + a² + a² + a² + a²) or by taking a big square (9a²) and subtracting three squares (9a² − 3a²). Both give 6a².

🧭 Overview

🧠 One-sentence thesis

Exponential properties allow us to simplify expressions by adding exponents when multiplying powers with the same base and by multiplying exponents when raising a power to another power.

📌 Key points (3–5)

• Product of powers rule: when multiplying powers with the same base, add the exponents (e.g., x² · x⁴ = x⁶).
• Power of a power rule: when raising a power to another power, multiply the exponents (e.g., (x²)³ = x⁶).
• Power of a product rule: when raising a product to a power, apply the exponent to each factor (e.g., (xy)² = x²y²).
• Common confusion: negative signs—distinguish between −5² (negative of 5 squared) and (−5)² (negative 5, squared); the first is negative, the second is positive.
• Why it matters: these properties simplify complex exponential expressions and are foundational for working with polynomials.

🔢 Basic exponential terminology

🔤 Base, exponent, and power

The excerpt uses a⁵ as an example to define key terms:

• Base: the number being multiplied repeatedly (in a⁵, the base is a).
• Exponent: the number indicating how many times to multiply the base (in a⁵, the exponent is 5).
• Power: the entire expression (a⁵ is called "a to the fifth power").
• Repeated multiplication: a⁵ means a · a · a · a · a.

➕➖ Sign rules for powers

The excerpt emphasizes distinguishing expressions by their signs:

• −(3⁴): the negative sign is outside; the result will be negative.
• −8²: means −(8²); the result is negative.
• (−5)²: the negative sign is inside the parentheses and is part of the base; squaring a negative gives a positive result.
• Don't confuse: −5² ≠ (−5)²; calculators give different answers because the first negates after squaring, the second squares the negative number.

🔗 Multiplying powers with the same base

➕ Product of powers property

When multiplying powers with the same base, add the exponents.

• The excerpt shows: 3² · 3⁷ and 2⁶ · 2.
• The rule: aᵐ · aⁿ = aᵐ⁺ⁿ.
• Example: x² · x⁴ = x²⁺⁴ = x⁶ (multiply x by itself 2 times, then 4 more times, for a total of 6 times).
• This works because repeated multiplication combines: (a · a) · (a · a · a · a) = a · a · a · a · a · a.

🔢 Writing in exponential notation

The excerpt asks students to rewrite repeated multiplication as powers:

• 2 · 2 = 2².
• (−3)(−3)(−3) = (−3)³.
• y · y · y · y · y = y⁵.
• (3a)(3a)(3a)(3a) = (3a)⁴.
• 6 · 6 · 6 · x · x · y · y · y · y = 6³ · x² · y⁴.

🔁 Raising powers to powers

✖️ Power of a power property

When raising a power to another power, multiply the exponents.

• The excerpt's Example 2: (x²)³ = x²·³ = x⁶.
• The rule: (aᵐ)ⁿ = aᵐⁿ.
• Why: (x²)³ means (x²)(x²)(x²) = x · x · x · x · x · x = x⁶.
• Example: (a³)⁴ = a³·⁴ = a¹².

🧮 Power of a product

When raising a product to a power, apply the exponent to each factor.

• The excerpt shows: (xy)² and (3a²b³)⁴.
• The rule: (ab)ⁿ = aⁿbⁿ.
• Example: (xy)² = x²y².
• More complex: (3a²b³)⁴ = 3⁴ · (a²)⁴ · (b³)⁴ = 81a⁸b¹².
• Example: (−2xy⁴z²)⁵ = (−2)⁵ · x⁵ · (y⁴)⁵ · (z²)⁵ = −32x⁵y²⁰z¹⁰.

🧩 Combining multiple properties

🔄 Multi-step simplification

The excerpt includes practice problems that combine several properties:

• (−2y⁴)(−3y): multiply coefficients and add exponents of y: (−2)(−3) · y⁴⁺¹ = 6y⁵.
• (4a²)(−3a)(−5a⁴): multiply coefficients: 4 · (−3) · (−5) = 60; add exponents: a²⁺¹⁺⁴ = a⁷; result is 60a⁷.
• (3x²y³) · (4xy²): multiply coefficients 3 · 4 = 12; add exponents for x: x²⁺¹ = x³; add exponents for y: y³⁺² = y⁵; result is 12x³y⁵.

🎯 Nested operations

When expressions involve both products and powers:

• (2a³b³)²: apply the power to each factor: 2² · (a³)² · (b³)² = 4a⁶b⁶.
• (−8x)³(5x)²: simplify each part first: (−8)³x³ · 5²x² = −512x³ · 25x² = −12,800x⁵.
• (4a²)(−2a³)⁴: simplify the power first: (−2)⁴(a³)⁴ = 16a¹²; then multiply: 4a² · 16a¹² = 64a¹⁴.

🔢 Special base cases

The excerpt includes problems with special bases:

• 1¹⁰ = 1: one raised to any power is one.
• 0³ = 0: zero raised to any positive power is zero.
• Decimal bases: (0.1)⁵ and (−0.6)³ follow the same rules; apply the exponent to the decimal.

🧭 Overview

🧠 One-sentence thesis

When multiplying binomials, certain patterns—squaring a binomial and multiplying a sum and difference—produce predictable results that simplify calculations and avoid common mistakes.

📌 Key points (3–5)

• General rule: multiply each term in the first binomial with each term in the second binomial, then combine like terms.
• Square of a binomial: produces three terms with a doubled middle term, following the pattern (a + b)² = a² + 2ab + b² or (a − b)² = a² − 2ab + b².
• Sum and difference pattern: multiplying (a + b)(a − b) makes the middle terms cancel, leaving only a² − b².
• Common confusion: (a + b)² does NOT equal a² + b²; the middle term 2ab is essential and cannot be omitted.
• Why it matters: these formulas speed up multiplication and apply to geometry problems and mental arithmetic.

🔢 General multiplication process

🔢 How to multiply two binomials

• Multiply each term in the first binomial with each term in the second binomial.
• Collect and combine like terms.
• Example: (2x + 3)(x + 4) expands to 2x² + 8x + 3x + 12, which simplifies to 2x² + 11x + 12.

🎯 What you get from two linear binomials

• Multiplying two linear binomials (degree 1) always produces a quadratic polynomial (degree 2).
• The result typically has four terms before simplification, then three terms (a trinomial) after combining like terms.

🟦 Squaring a binomial

🟦 What "squaring" means

Raising a polynomial to a power means multiplying the polynomial by itself however many times the exponent indicates.

• (x + 4)² is the same as (x + 4)(x + 4).
• When you expand it, you get x² + 4x + 4x + 16, which simplifies to x² + 8x + 16.
• Notice the two middle terms are identical and add together.

📐 The general pattern for squaring

The excerpt derives the pattern by squaring a general binomial:

• (a + b)² = a² + 2ab + b²
• (a − b)² = a² − 2ab + b²

How to remember it:

• Square the first term.
• Add or subtract twice the product of the two terms.
• Add the square of the second term.

⚠️ Common mistake to avoid

• Don't confuse: (a + b)² does NOT equal a² + b².
• The middle term 2ab is required to make the equation work.
• Example: if a = 4 and b = 3, then (4 + 3)² = 49, but 4² + 3² = 25. They are not equal.

🧮 Examples of squaring binomials

The excerpt provides four worked examples:

Expression	Let a =	Let b =	Result
(x + 10)²	x	10	x² + 20x + 100
(2x − 3)²	2x	3	4x² − 12x + 9
(x² + 4)²	x²	4	x⁴ + 8x² + 16
(5x − 2y)²	5x	2y	25x² − 20xy + 4y²

• Each example applies the formula: square the first term, double the product, square the second term.

➕➖ Sum and difference pattern

➕➖ What happens when you multiply (a + b)(a − b)

• Example: (x + 4)(x − 4) = x² − 4x + 4x − 16 = x² − 16.
• The middle terms are opposites of each other, so they cancel out when you collect like terms.

🔑 The sum and difference formula

(a + b)(a − b) = a² − b²

• When multiplying a sum and difference of the same two terms, the middle terms always cancel.
• You get the square of the first term minus the square of the second term.

🧮 Examples of sum and difference

The excerpt provides four worked examples:

Expression	Let a =	Let b =	Result
(x + 3)(x − 3)	x	3	x² − 9
(5x + 9)(5x − 9)	5x	9	25x² − 81
(2x³ + 7)(2x³ − 7)	2x³	7	4x⁶ − 49
(4x + 5y)(4x − 5y)	4x	5y	16x² − 25y²

• Each example shows that only two terms remain after the middle terms cancel.

🌍 Real-world applications

🌍 Where these patterns are used

• Geometry problems: the excerpt mentions finding the area of a square with side length (a + b), which uses the squaring formula.
• Mental arithmetic: the formulas speed up calculations without writing out every step.
• The excerpt begins an example about finding the area of a square but does not complete it in the provided text.

🧭 Overview

🧠 One-sentence thesis

Special product formulas—the sum-and-difference formula and the square-of-a-binomial formula—allow us to multiply certain polynomial pairs quickly by recognizing patterns that eliminate middle terms or group like terms predictably.

📌 Key points (3–5)

• Sum and difference formula: multiplying (a + b)(a − b) always gives a² − b², because the middle terms cancel out.
• Why middle terms cancel: when you expand (a + b)(a − b), you get −ab and +ab, which are opposites and sum to zero.
• Square of a binomial: squaring (a + b) gives a² + 2ab + b², which can be visualized geometrically as the area of a square with side length (a + b).
• Common confusion: don't confuse (a + b)(a − b) with (a + b)² — the first has opposite signs and cancels middle terms; the second has the same sign and doubles the middle term.
• Real-world use: these formulas speed up mental arithmetic and solve geometry problems involving areas.

🔄 The sum-and-difference formula

🔄 What happens when you multiply a sum and a difference

Sum and Difference Formula: (a + b)(a − b) = a² − b²

• When you multiply two binomials that have the same two terms but opposite signs in the middle, the middle terms always cancel.
• Expanding (a + b)(a − b) step-by-step:
  • (a + b)(a − b) = a² − ab + ab − b²
  • The −ab and +ab are opposites, so they sum to zero.
  • Result: a² − b²
• This is not a coincidence; it always happens with this pattern.

🧮 Examples of sum-and-difference products

The excerpt provides four worked examples:

Expression	Identify a and b	Apply formula	Result
(x + 3)(x − 3)	a = x, b = 3	x² − 3²	x² − 9
(5x + 9)(5x − 9)	a = 5x, b = 9	(5x)² − 9²	25x² − 81
(2x³ + 7)(2x³ − 7)	a = 2x³, b = 7	(2x³)² − 7²	4x⁶ − 49
(4x + 5y)(4x − 5y)	a = 4x, b = 5y	(4x)² − (5y)²	16x² − 25y²

• Key: recognize the pattern (same two terms, opposite signs) and apply the formula directly.
• Don't confuse: (a + b)(a − b) ≠ (a + b)² — the first cancels middle terms; the second does not.

🟦 The square-of-a-binomial formula

🟦 What happens when you square a binomial

• Squaring (a + b) means multiplying (a + b)(a + b).
• The result is a² + 2ab + b².
• The middle term is doubled (2ab), not canceled.

🖼️ Geometric visualization

The excerpt explains the square-of-a-binomial formula using a square with side length (a + b):

• The total area is (a + b)(a + b).
• Break the square into four regions:
  • One blue square with area a²
  • One red square with area b²
  • Two rectangles, each with area ab
• Adding all regions: a² + ab + ab + b² = a² + 2ab + b²

This visual proof shows why the middle term is 2ab: there are two rectangles of area ab.

🧮 Real-world applications

🧮 Mental arithmetic shortcuts

The excerpt shows how to use special product formulas to multiply numbers quickly without a calculator.

Strategy: rewrite each number as a sum or difference of numbers that are easy to square.

📐 Using sum-and-difference for multiplication

• Example: 43 × 57
  • Rewrite: 43 = (50 − 7), 57 = (50 + 7)
  • Apply formula: (50 − 7)(50 + 7) = 50² − 7² = 2500 − 49 = 2451
• Example: 112 × 88
  • Rewrite: 112 = (100 + 12), 88 = (100 − 12)
  • Apply formula: (100 + 12)(100 − 12) = 100² − 12² = 10000 − 144 = 9856

📐 Using square-of-a-binomial for squaring

• Example: 45²
  • Rewrite: 45 = (40 + 5)
  • Apply formula: (40 + 5)² = 40² + 2(40)(5) + 5² = 1600 + 400 + 25 = 2025

📐 Nested application

• Example: 481 × 319
  • Rewrite: 481 = (400 + 81), 319 = (400 − 81)
  • Apply sum-and-difference: (400 + 81)(400 − 81) = 400² − 81²
  • 400² = 160000 (easy)
  • 81² is harder, so rewrite 81 = (80 + 1) and apply square-of-a-binomial:
    • 81² = (80 + 1)² = 80² + 2(80)(1) + 1² = 6400 + 160 + 1 = 6561
  • Final result: 160000 − 6561 = 153439

🏗️ Geometry problems

The excerpt mentions using special products to find areas of squares.

• Example: a square with side length (a + b) has area (a + b)(a + b) = a² + 2ab + b².
• This connects the algebraic formula to a concrete geometric interpretation.

🔍 Common confusions

🔍 Sum-and-difference vs square-of-a-binomial

Pattern	Formula	Middle terms
(a + b)(a − b)	a² − b²	Cancel out (−ab and +ab sum to zero)
(a + b)(a + b) or (a + b)²	a² + 2ab + b²	Double (two rectangles of area ab)

• Don't confuse: opposite signs in the factors → middle terms cancel; same signs → middle terms double.
• The excerpt emphasizes that the cancellation in (a + b)(a − b) is not a coincidence—it always happens.

🔍 Identifying a and b correctly

• When applying the formulas, correctly identify which parts correspond to a and which to b.
• Example: in (5x + 9)(5x − 9), a = 5x (not just x) and b = 9.
• Squaring a multi-term expression: (5x)² = 25x², not 5x².

🧭 Overview

🧠 One-sentence thesis

When dividing exponential expressions with the same base, you keep the base and subtract the exponents, and when raising a quotient to a power, you apply that power to both the numerator and denominator.

📌 Key points (3–5)

• Quotient of Powers Property: When dividing same bases, subtract the bottom exponent from the top exponent (x to the n divided by x to the m equals x to the n minus m).
• Power of a Quotient Property: When raising a fraction to a power, multiply the power outside the parentheses with both the numerator and denominator powers inside.
• When to apply each rule: Use quotient of powers for division of same bases; use power of a quotient when an entire fraction is raised to a power.
• Common confusion: The quotient rule only works when the bases are the same; for different bases, apply the rule separately to each base.
• Why it matters: These properties allow you to simplify complex exponential expressions systematically without writing out repeated multiplication.

📐 The Quotient of Powers Property

📐 What the property states

Quotient of Powers Property: When the exponent in the numerator is larger than the exponent in the denominator, for all real numbers x, x to the n divided by x to the m equals x to the n minus m.

• The key action: keep the base and subtract exponents.
• Subtract the denominator (bottom) exponent from the numerator (top) exponent.
• This only applies when the bases are the same.

🔍 Why this property works

The excerpt shows the reasoning through repeated multiplication:

• x to the 7 divided by x to the 4 can be written as (x · x · x · x · x · x · x) divided by (x · x · x · x).
• Four x's cancel out from top and bottom.
• This leaves x · x · x over 1, which equals x to the 3.
• The shortcut: 7 minus 4 equals 3, so x to the 7 divided by x to the 4 equals x to the 3.

🧮 Multiple bases in one expression

When you have different bases, apply the rule separately for each base.

Example from the excerpt:

• x to the 5 times y to the 3, all divided by x to the 3 times y to the 2
• Apply to x: x to the (5 minus 3) equals x to the 2
• Apply to y: y to the (3 minus 2) equals y to the 1, or just y
• Final answer: x to the 2 times y

Don't confuse: You cannot combine or subtract exponents across different bases—only same bases can use this property.

🎯 The Power of a Quotient Property

🎯 What the property states

Power of a Quotient Property: (x to the n divided by y to the m) all raised to the p equals x to the (n times p) divided by y to the (m times p).

• The power outside the parentheses multiplies with the powers inside.
• This applies to both the numerator and the denominator.

🔍 Why this property works

The excerpt demonstrates with repeated multiplication:

• (x to the 3 divided by y to the 2) all raised to the 4 means multiplying the fraction by itself 4 times.
• This gives (x · x · x) · (x · x · x) · (x · x · x) · (x · x · x) on top.
• And (y · y) · (y · y) · (y · y) · (y · y) on bottom.
• Counting: 12 x's on top (3 times 4) and 8 y's on bottom (2 times 4).
• Result: x to the 12 divided by y to the 8.

🧮 Applying the shortcut

Example from the excerpt:

• (x to the 10 divided by y to the 5) all raised to the 3
• Multiply 10 times 3 for the numerator: x to the 30
• Multiply 5 times 3 for the denominator: y to the 15
• Final answer: x to the 30 divided by y to the 15

🔧 Working with both properties together

🔧 Combining the rules

Some problems require using both properties in sequence:

• First, simplify any powers of quotients (multiply exponents).
• Then, apply the quotient of powers rule (subtract exponents) if needed.

⚠️ Common pitfalls

Mistake	Correct approach
Adding exponents when dividing	Subtract exponents when dividing same bases
Applying quotient rule to different bases	Only apply to matching bases; handle each base separately
Forgetting to apply power to both parts	Power of a quotient applies to numerator AND denominator

Don't confuse: Division of powers (subtract exponents) versus power of a quotient (multiply exponents)—these are different operations for different situations.

🧭 Overview

🧠 One-sentence thesis

Negative exponents allow us to rewrite fractions as expressions without denominators, and applying exponent rules correctly requires careful attention to which terms the exponent affects and to the order of operations.

📌 Key points (3–5)

• Negative exponents eliminate fractions: expressions like 1/x² can be rewritten as x⁻².
• Apply negative exponents to each variable separately: when multiple variables appear in a denominator, each gets its own negative exponent.
• Common confusion: when a constant is in the numerator with a variable in the denominator (like 2/x²), only the variable term gets the negative exponent, resulting in 2x⁻².
• Order of operations matters: always evaluate parentheses first, then exponents, then multiplication/division, then addition/subtraction.
• Final answers use positive exponents: simplification problems require converting all negative exponents back to positive form.

✏️ Converting fractions to negative exponents

✏️ Basic negative exponent rule

The negative power rule: a term in the denominator can be moved to the numerator by changing the sign of its exponent.

• 1/x² becomes x⁻²
• The reciprocal relationship: moving between numerator and denominator flips the exponent sign
• Example: 2/x² = 2x⁻² (the constant 2 stays in the numerator, only x² becomes x⁻²)

🔢 Multiple variables

When a fraction has multiple variables in the denominator, each variable is handled independently:

• x²/y³ becomes x²y⁻³
• Each variable in the denominator gets its own negative exponent
• Variables already in the numerator keep their positive exponents
• Example: 3/xy can be written as 3x⁻¹y⁻¹

⚠️ Constants vs variables

Don't confuse: A constant in the numerator does not get a negative exponent.

• In 2/x², the 2 remains as a coefficient: 2x⁻²
• Think of it as 2 · (1/x²) = 2 · x⁻²
• Only the variable terms in the denominator become negative exponents

🧮 Order of operations with exponents

🧮 The correct sequence

The excerpt emphasizes following this order:

1. Evaluate inside parentheses or grouping symbols
2. Evaluate exponents
3. Perform multiplication/division from left to right
4. Perform addition/subtraction from left to right

📝 Worked example from the excerpt

Problem: 3 · 5² − 10 · 5 + 1

Step-by-step:

• First, evaluate the exponent: 5² = 25
• Result: 3 · 25 − 10 · 5 + 1
• Next, multiply: 3 · 25 = 75 and 10 · 5 = 50
• Result: 75 − 50 + 1
• Finally, subtract and add: 75 − 50 = 25, then 25 + 1 = 26

Don't confuse: Do not multiply before evaluating exponents; exponents come before multiplication in the order of operations.

🔄 Simplifying to positive exponents only

🔄 The final form requirement

The excerpt's practice problems specify: "Be sure the final answer includes only positive exponents."

• Negative exponents are useful for eliminating fractions during work
• Final answers should convert back to positive exponents
• This means moving terms with negative exponents back to the denominator

🔄 Example patterns

Expression with negative exponents	Expression with positive exponents only
x⁻¹ · y²	y²/x
2x⁻²	2/x²
a²b⁻³c⁻¹	a²/(b³c)

🧭 Overview

🧠 One-sentence thesis

Dividing a polynomial by a monomial requires dividing each term in the numerator separately by the common denominator, and the most common error is canceling the denominator with only one term instead of all terms.

📌 Key points (3–5)

• Core method: The denominator serves as the common divisor for all terms in the numerator—split the fraction and divide each term separately.
• Sign handling: A negative denominator changes the sign of every resulting fraction.
• Common confusion: Students often cancel the denominator with just one term in the numerator instead of dividing every term; for example, (3x + 4) divided by 4 is NOT 3x, but rather (3x/4) + 1.
• Simplification steps: Write each term over the denominator, reduce each fraction individually, then combine the results.

➗ The core division method

➗ How to divide a polynomial by a monomial

The bottom of the fraction serves as the common denominator to all the terms in the numerator.

• What this means: When you have a polynomial (multiple terms) divided by a single term (monomial), you must divide each term in the numerator by the denominator.
• Why: The denominator applies to the entire numerator, not just part of it.
• Process:
  1. Write each term in the numerator as a separate fraction with the same denominator.
  2. Simplify each fraction individually.
  3. Combine the simplified terms.

📝 Step-by-step breakdown

Example: (8x squared minus 4x plus 16) divided by 2

• Split into three fractions: (8x squared)/2 minus (4x)/2 plus 16/2
• Simplify each: 4x squared minus 2x plus 8
• Result: three separate terms, each simplified

Example: (3x cubed plus 6x minus 1) divided by x

• Split: (3x cubed)/x plus (6x)/x minus 1/x
• Simplify: 3x squared plus 6 minus (1/x)
• Notice the last term remains a fraction because 1 and x do not divide evenly.

⚠️ Common errors and sign rules

⚠️ The "canceling only one term" mistake

The excerpt emphasizes:

A common error is to cancel the denominator with just one term in the numerator.

• The mistake: For (3x + 4) divided by 4, students incorrectly cross out the 4 in the numerator and denominator, leaving just 3x.
• Why it's wrong: The denominator of 4 is common to both terms in the numerator—you are dividing both 3x and 4 by 4.
• Correct approach: (3x + 4)/4 = (3x)/4 + 4/4 = (3x)/4 + 1

Don't confuse: Canceling works only when the entire numerator is a single term that shares a factor with the denominator; when the numerator has multiple terms, you must split first.

➖ Handling negative denominators

Example: (5x cubed minus 10x squared plus x minus 25) divided by (negative 5x squared)

• Key rule: The negative sign in the denominator changes all the signs of the fractions.
• Split into fractions: (5x cubed)/(−5x squared) minus (10x squared)/(−5x squared) plus x/(−5x squared) minus 25/(−5x squared)
• Rewrite with negatives distributed: −(5x cubed)/(5x squared) + (10x squared)/(5x squared) − x/(5x squared) + 25/(5x squared)
• Simplify: −x + 2 − 1/(5x) + 5/(x squared)

Why this matters: Forgetting to apply the negative sign to every term leads to incorrect signs in the final answer.

🧮 Worked examples

🧮 Example with all positive terms

Divide (−3x squared minus 18x plus 6) by (9x):

Step	Work	Result
Split	(−3x squared)/(9x) − (18x)/(9x) + 6/(9x)	Three separate fractions
Simplify each	−(x/3) − 2 + 2/(3x)	Each term reduced
Final form	−x/3 − 2 + 2/(3x)	Combined expression

🧮 Example with mixed terms

The excerpt shows that when dividing (x cubed minus 12x squared plus 3x minus 4) by (12x squared):

• Each term becomes a fraction over 12x squared.
• Some terms simplify to whole expressions (like x cubed over 12x squared becomes x/12).
• Some remain as fractions (like 4 over 12x squared becomes 1 over 3x squared).

Key takeaway: Not every term will simplify to a polynomial term; some will remain as rational expressions (fractions with variables in the denominator).

📋 Practice structure

The excerpt includes 11 practice problems covering:

• Simple monomials divided by constants (e.g., (2x + 4)/2)
• Polynomials divided by variables (e.g., (x squared + 2x − 5)/x)
• Negative denominators (e.g., (4x squared + 12x − 36)/(−4x))
• Higher-degree polynomials (e.g., (x cubed − 12x squared + 3x − 4)/(12x squared))
• Denominators with higher powers (e.g., (x squared − 6x − 12)/(5x to the fourth))

Purpose: These problems reinforce the core method—split every term, simplify each fraction, watch for negative signs, and combine results.

🧭 Overview

🧠 One-sentence thesis

An equation's solution is the value that makes both sides equal, and checking solutions by substitution confirms whether a proposed value truly satisfies the equation.

📌 Key points (3–5)

• What an equation is: a mathematical sentence connecting an expression to a value, variable, or another expression with an equal sign.
• What a solution means: the value (or multiple values) that make the equation true when substituted for the variable.
• How to check a solution: substitute the proposed value for the variable and verify that both sides of the equation balance (produce a true statement).
• Common confusion: a value is only a solution if substitution produces a true statement like 17 = 17; if you get something false like 15 = 8, the value is NOT a solution.

🔍 What equations and solutions are

🔍 Definition of an equation

An algebraic equation is a mathematical sentence connecting an expression to a value, variable, or another expression with an equal sign (=).

• An equation is formed when an algebraic expression is set equal to another value, variable, or expression.
• The equal sign (=) is what makes it an equation rather than just an expression.
• Example: "1/4 times m equals 20.00" is an equation; "3x + 2" alone is just an expression.

🎯 Definition of a solution

The solution to an equation is the value (or multiple values) that make the equation true.

• A solution is not just any number—it must make the equation balance when substituted in.
• There can be one solution, multiple solutions, or no solution depending on the equation.
• Example: In the equation "one-quarter m = 20.00," the solution is 80 because one-quarter of 80 equals 20.00.

✅ How to check a solution

✅ The substitution method

• To verify a proposed solution, substitute the value for the variable in the original equation.
• Calculate both sides separately.
• If both sides produce the same number, the equation is true and the value is a solution.
• If both sides produce different numbers, the equation is false and the value is NOT a solution.

📝 Example: Confirming a solution

Checking x = 5 in the equation 3x + 2 = -2x + 27:

• Substitute 5 for x on both sides:
  • Left side: 3 times 5 plus 2 equals 15 plus 2 equals 17
  • Right side: negative 2 times 5 plus 27 equals negative 10 plus 27 equals 17
• Because 17 = 17 is a true statement, x = 5 is a solution.

Don't confuse: Just because you can substitute a number doesn't mean it's a solution—the final statement must be true.

📝 Example: Rejecting a non-solution

Checking z = 3 in the equation z squared plus 2z = 8:

• Substitute 3 for z:
  • Left side: 3 squared plus 2 times 3 equals 9 plus 6 equals 15
  • Right side: 8
• Because 15 = 8 is NOT a true statement, z = 3 is not a solution.

🧮 Why checking matters

🧮 Verification ensures correctness

• The excerpt emphasizes that "checking an answer to an equation is almost as important as the equation itself."
• Substitution confirms that both sides of the equation balance.
• Without checking, you cannot be certain that your proposed solution is correct.

🧮 Real-world context

• The excerpt mentions that equations are used in many career fields:
  • Medical researchers use equations to determine drug circulation time.
  • Botanists use equations to determine tree growth time.
  • Environmental scientists use equations to approximate species repopulation time.
• In these contexts, an incorrect solution could lead to wrong predictions or decisions, so verification is essential.

🧭 Overview

🧠 One-sentence thesis

Solving one-step equations means isolating the variable by applying a single inverse operation to both sides of the equation, using either the Addition Property of Equality or the Multiplication Property of Equality.

📌 Key points (3–5)

• What solving means: writing an equivalent equation with the variable isolated on one side.
• Inverse operations: each operation (addition, subtraction, multiplication, division, exponents, roots) has an inverse that "undoes" it.
• Two key properties: the Addition Property of Equality and the Multiplication Property of Equality allow you to perform the same operation on both sides of an equation.
• Common confusion: when dividing by a fraction, you actually multiply by its reciprocal (e.g., to cancel one-eighth, multiply by 8, not divide by one-eighth).
• One-step equations: equations that require exactly one operation to isolate the variable.

🎯 What it means to solve an equation

🎯 Isolating the variable

To solve an equation means to write an equivalent equation that has the variable by itself on one side. This is also known as isolating the variable.

• The goal is not just to find a number, but to rewrite the equation so the variable stands alone.
• Example: if you have "16 = y − 11", you want to end up with "y = (some number)".

🔄 Equivalent equations

Equivalent equations are two or more equations having the same solution.

• When you apply inverse operations to both sides, you create a new equation that looks different but has the same solution.
• Example: "16 = y − 11" and "27 = y" are equivalent because y = 27 satisfies both.

✅ Checking solutions

• Substitute the value back into the original equation and verify both sides balance.
• Example: to check that x = 5 solves "3x + 2 = −2x + 27", substitute 5 for x: "3·5 + 2 = −2·5 + 27" becomes "17 = 17", which is true.
• If the statement is false (like "15 = 8"), the value is not a solution.

➕ Solving with addition or subtraction

🔑 The Addition Property of Equality

For all real numbers a, b, and c: If a = b, then a + c = b + c.

• What you do to one side of an equation, you must do to the other to maintain equality.
• This property also works for subtraction (since subtracting is adding a negative).

🧮 How to solve using addition or subtraction

• Identify the operation attached to the variable.
• Apply the inverse operation to both sides.
• Example: "16 = y − 11"
  • The variable y has 11 subtracted from it.
  • Add 11 to both sides: "16 + 11 = y − 11 + 11"
  • Simplify: "27 = y"

🐴 Real-world scenario

Example: A Shetland pony is loaded onto a trailer that weighs 2,200 pounds empty. The new weight is 2,550 pounds. How much does the pony weigh?

• Let p = weight of the pony.
• Equation: "2550 = 2200 + p"
• Subtract 2200 from both sides: "2550 − 2200 = p"
• Solution: p = 350 pounds.

✖️ Solving with multiplication or division

🔑 The Multiplication Property of Equality

For all real numbers a, b, and c: If a = b, then a(c) = b(c).

• You can multiply (or divide) both sides by the same non-zero number and maintain equality.
• Division is like multiplying by the reciprocal, so this property covers both operations.

🧮 How to solve using multiplication or division

• If the variable is multiplied by a number, divide both sides by that number.
• If the variable is divided by a number, multiply both sides by that number.
• Example: "−8k = −96"
  • The variable k is multiplied by −8.
  • Divide both sides by −8: "−8k ÷ −8 = −96 ÷ −8"
  • Solution: k = 12.

🍰 Working with fractions

• Remember: (a/b) × (b/a) = 1.
• To cancel a fraction, multiply by its reciprocal (not divide by the fraction).
• Example: "one-eighth · x = 1.5"
  • Multiply both sides by 8 (the reciprocal of one-eighth): "8 · (one-eighth · x) = 8 · 1.5"
  • Solution: x = 12.
• Don't confuse: dividing by one-eighth is not the same as multiplying by 8; multiplying by the reciprocal is the correct approach.

🌍 Real-world applications

🌱 Growth and time problems

Example: Tomato seeds grow into plants and bear ripe fruit in 19 weeks. Lorna planted her seeds 11 weeks ago. How long must she wait?

• Let w = weeks until tomatoes are ready.
• Equation: "w + 11 = 19"
• Subtract 11 from both sides: "w = 8"
• She must wait 8 more weeks.

🌭 Rate problems

Example: Takeru Kobayashi ate 53.5 hot dogs in 12 minutes. How many minutes did it take to eat one hot dog?

• Let m = minutes per hot dog.
• Equation: "53.5m = 12"
• Divide both sides by 53.5: "m = 12 ÷ 53.5"
• Solution: m ≈ 0.224 minutes (or about 13.5 seconds per hot dog).

🔧 Inverse operations reference

Operation	Inverse Operation	Example
Addition	Subtraction	To undo "+ 5", subtract 5
Subtraction	Addition	To undo "− 3", add 3
Multiplication	Division	To undo "× 4", divide by 4
Division	Multiplication	To undo "÷ 2", multiply by 2
Exponent	Root	To undo squaring, take square root

• Inverse operations "undo" each other when combined.
• Use the appropriate inverse to isolate the variable in one step.

🧭 Overview

🧠 One-sentence thesis

Two-step equations require undoing two operations—typically addition/subtraction followed by multiplication/division—to isolate the variable and find its value.

📌 Key points (3–5)

• What two-step equations are: equations of the form ax + b = some number, requiring two operations to solve.
• Solution procedure: first use the Addition Property of Equality to isolate the variable term (ax), then use the Multiplication Property of Equality to isolate the variable (x).
• Combining like terms: when multiple terms share identical variable parts, add or subtract their coefficients before solving.
• Common confusion: like terms can only be combined through addition/subtraction, not multiplication/division.
• Real-world applications: two-step equations model situations involving rates, taxes, service charges, and conversions.

🔧 The Two-Step Solution Process

🔧 Core procedure for ax + b = some number

The excerpt provides a systematic approach:

Step 1: Use the Addition Property of Equality to isolate the variable term

• Goal: get ax alone on one side
• Method: add or subtract the constant term (b) from both sides
• Result: ax = some number

Step 2: Use the Multiplication Property of Equality to isolate the variable

• Goal: get x alone on one side
• Method: divide or multiply both sides by the coefficient (a)
• Result: x = some number

📝 Weight loss example

Example: Shaun weighs 146 pounds and wants to reach 130 pounds by losing 2 pounds per week. How many weeks will this take?

Equation: -2w + 146 = 130

• Apply Addition Property: -2w + 146 - 146 = 130 - 146
• Simplify: -2w = -16
• Apply Multiplication Property: -2w ÷ -2 = -16 ÷ -2
• Solution: w = 8 weeks

🧮 Combining Like Terms

🧮 What like terms are

Like terms: expressions that have identical variable parts.

Key rules from the excerpt:

• You can only combine like terms if the variable parts are identical
• Combining like terms applies only to addition and subtraction
• This is NOT true for multiplication and division

🔢 The coefficient concept

Coefficient: the numerical part of an algebraic term.

• To combine like terms, add or subtract the coefficients
• Remember that a variable alone (p) equals 1 times that variable (1p)
• Example: p + 0.06p = 1p + 0.06p = 1.06p

💰 Sales tax example

Example: A purchase with 6% sales tax costs $95.12 total. What was the price before tax?

Equation: p + 0.06p = 95.12

• Combine like terms: 1.06p = 95.12
• Apply Multiplication Property: 1.06p ÷ 1.06 = 95.12 ÷ 1.06
• Solution: p = $89.74

🌍 Real-World Applications

🔧 Service charges

Example: A plumber charges $65 call-out fee plus $75 per hour. Total bill is $196.25. How long did the job take?

Equation: 65 + 75(h) = 196.25

• Apply Addition Property: 65 + 75(h) - 65 = 196.25 - 65
• Simplify: 75(h) = 131.25
• Apply Multiplication Property: h = 1.75 hours (1 hour, 45 minutes)

🌡️ Temperature conversion

Example: Convert 89°F to Celsius using the formula: multiply Celsius by 1.8 then add 32.

Equation: 1.8C + 32 = 89

• Apply Addition Property: 1.8C + 32 - 32 = 89 - 32
• Simplify: 1.8C = 57
• Apply Multiplication Property: 1.8C ÷ 1.8 = 57 ÷ 1.8
• Solution: C = 31.67°C

🛒 Budgeting scenarios

The excerpt includes several practical problems:

Scenario	Unknown variable	Equation structure
Sales tax calculation	Original price	price + (tax rate)(price) = total
Service with hourly rate	Time worked	flat fee + (hourly rate)(hours) = total
Temperature conversion	Celsius value	(conversion factor)(C) + constant = Fahrenheit

⚠️ Key Distinctions

⚠️ When to combine vs when to distribute

• Combine like terms: only when terms have identical variable parts and are connected by addition/subtraction
• Don't confuse: multiplication and division do NOT follow the same combining rules
• The excerpt emphasizes: "Combining like terms only applies to addition and subtraction!"

⚠️ Order of operations in solving

• Always isolate the variable term (ax) before isolating the variable (x)
• First undo addition/subtraction, then undo multiplication/division
• This reverses the order in which operations were originally applied

🧭 Overview

🧠 One-sentence thesis

Multi-step equations require a systematic procedure—combining like terms, applying the Distributive Property, and isolating the variable through multiple operations—to find the solution.

📌 Key points (3–5)

• What multi-step equations are: equations that require more than two operations to solve, including combining like terms or using the Distributive Property.
• The standard procedure: remove parentheses first, combine like terms, isolate the variable term, then isolate the variable itself, and always check your solution.
• Two ways to handle parentheses: use the Distributive Property to expand, or use the Multiplication Property of Equality to remove them.
• Common confusion: don't skip steps—follow the procedure in order (parentheses → like terms → isolate term → isolate variable).
• Why it matters: real-world problems often involve multiple quantities and operations, requiring multi-step solving techniques.

🔧 The standard solving procedure

🔧 Five-step process

The excerpt provides a systematic procedure to solve any equation:

Procedure to Solve Equations:

1. Remove any parentheses by using the Distributive Property or the Multiplication Property of Equality
2. Simplify each side of the equation by combining like terms
3. Isolate the ax term. Use the Addition Property of Equality to get the variable on one side of the equal sign and the numerical values on the other
4. Isolate the variable. Use the Multiplication Property of Equality to get the variable alone on one side of the equation
5. Check your solution

• This procedure applies to all equation types, not just multi-step equations.
• Each step builds on the previous one—order matters.
• The final check step confirms whether your solution is correct.

⚠️ Why order matters

• You cannot isolate the variable if like terms are still scattered.
• You cannot combine like terms accurately if parentheses are still present.
• Following the procedure prevents errors and missed steps.

🧮 Solving by combining like terms

🧮 What like terms are

• Like terms are terms that contain the same variable to the same power.
• Example from the excerpt: 3p + 4p are like terms because both contain the variable p.
• Combining them simplifies the equation: 3p + 4p = 7p.

🎃 Party planning example

The excerpt uses a Halloween party scenario:

• Setup: 3 cans of soda per person, 4 slices of pizza per person, and 37 party favors total 79 items.

• Equation: 3p + 4p + 37 = 79

• Solution steps:

  1. Combine like terms: 7p + 37 = 79
  2. Apply Addition Property of Equality: 7p + 37 - 37 = 79 - 37
  3. Simplify: 7p = 42
  4. Apply Multiplication Property of Equality: 7p ÷ 7 = 42 ÷ 7
  5. Solution: p = 6 (six people are coming)

• Why this is multi-step: it requires combining terms, then subtracting, then dividing—three distinct operations.

📦 Solving with the Distributive Property

📦 When to use the Distributive Property

• When an equation has parentheses with a coefficient outside, such as 2(5x + 9) = 78.
• The Distributive Property removes parentheses by multiplying the outside term by each term inside.
• Example: 2(5x + 9) becomes 10x + 18.

🔢 Worked example

The excerpt shows how to solve 2(5x + 9) = 78:

1. Apply Distributive Property: 10x + 18 = 78
2. Apply Addition Property of Equality: 10x + 18 - 18 = 78 - 18
3. Simplify: 10x = 60
4. Apply Multiplication Property of Equality: 10x ÷ 10 = 60 ÷ 10
5. Solution: x = 6
6. Check: Does 10(6) + 18 = 78? Yes, so the answer is correct.

🐕 Puppy enclosure example

Kashmir wants to fence three sides of a puppy run (connecting to his porch):

• Given: the run is 12 feet long, and he has 40 feet of fencing.
• Equation: w + w + 12 = 40 (two widths plus one length)
• Solution steps:
  1. Combine like terms: 2w + 12 = 40
  2. Subtract 12: 2w = 28
  3. Divide by 2: w = 14
• Result: the enclosure is 14 feet wide by 12 feet long.

🔄 Don't confuse: Distributive Property vs. Multiplication Property

• Distributive Property: expands a(b + c) into ab + ac.
• Multiplication Property of Equality: multiplies or divides both sides of an equation by the same number.
• The excerpt mentions both can remove parentheses, but this lesson focuses on the Distributive Property approach.

🎯 Key strategies and checks

🎯 Always check your solution

• Substitute your answer back into the original equation.
• Example: for x = 6 in 2(5x + 9) = 78, check if 10(6) + 18 = 78. It does, so the solution is correct.
• This step catches arithmetic mistakes.

🧩 Breaking down complex problems

• Translate word problems into algebraic equations first.
• Identify what the variable represents (e.g., number of people, width, number of weeks).
• Use the five-step procedure systematically.

📋 Summary table

Step	What to do	Example operation
1. Remove parentheses	Use Distributive Property	2(5x + 9) → 10x + 18
2. Combine like terms	Add/subtract like terms	3p + 4p → 7p
3. Isolate variable term	Use Addition Property	7p + 37 = 79 → 7p = 42
4. Isolate variable	Use Multiplication Property	7p = 42 → p = 6
5. Check	Substitute back	Does 7(6) + 37 = 79? Yes.

• Following this order prevents confusion and ensures all operations are applied correctly.

🧭 Overview

🧠 One-sentence thesis

Equations with variables on both sides require using the Addition Property of Equality to gather all variable terms on one side before isolating the variable and solving.

📌 Key points (3–5)

• What makes these equations different: the variable appears on both sides of the equal sign, not just one.
• Core solving strategy: use the Addition Property of Equality to move all variable terms to one side and all numerical values to the other.
• Step sequence: gather variables on one side → isolate the variable → check your solution.
• Common confusion: don't forget to apply the Distributive Property first if parentheses are present before gathering variables.
• Why it matters: real-world problems (like comparing bank account growth) often produce equations with variables on both sides.

🔧 The solving process

🔧 Gathering variables on one side

• The key step is using the Addition Property of Equality to collect all terms containing the variable on one side of the equation.
• You subtract (or add) the same variable term from both sides to eliminate it from one side.
• Example: In 125 + 20w = 43 + 37w, subtract 20w from both sides to get 125 = 43 + 17w.
• Don't confuse: this is not "moving" terms—you are adding or subtracting the same amount from both sides to maintain equality.

🎯 Isolating the variable

• Once variables are on one side and numbers on the other, use the same techniques from earlier lessons.
• Combine like terms, then use inverse operations (addition/subtraction, then multiplication/division) to isolate the variable.
• Example: From 125 = 43 + 17w, subtract 43 to get 82 = 17w, then divide by 17 to get w ≈ 4.82.

✅ Checking your solution

• The excerpt lists "Check your solution" as the final step in the process.
• Substitute your answer back into the original equation to verify both sides are equal.

🧮 Working with parentheses

🧮 Distributive Property first

• When parentheses appear in an equation with variables on both sides, you must remove them before gathering variables.
• Use the Distributive Property to expand expressions like 3(h + 1) into 3h + 3.
• Example: 3(h + 1) = 11h - 23 becomes 3h + 3 = 11h - 23 after distributing.

🧮 Then gather and solve

• After removing parentheses, follow the standard process: gather variables, isolate, solve.
• Example: From 3h + 3 = 11h - 23, subtract 3h from both sides to get 3 = 8h - 23, then add 23 to get 26 = 8h, finally divide by 8 to get h = 3.25.

📖 Real-world application example

📖 The bank account problem

• Scenario: Karen starts with $125 and deposits $20 each week; Sarah starts with $43 and deposits $37 each week. When will they have the same amount?
• Translation to equation: Let w = number of weeks. Karen's amount: 125 + 20w. Sarah's amount: 43 + 37w. Set them equal: 125 + 20w = 43 + 37w.
• Why variables appear on both sides: each person's account balance depends on the same variable (weeks), creating variable terms on both sides of the equation.

📖 Solving the bank account problem

Step	Action	Result
1. Write equation	Translate the situation	125 + 20w = 43 + 37w
2. Gather variables	Subtract 20w from both sides	125 = 43 + 17w
3. Isolate variable	Subtract 43, then divide by 17	w ≈ 4.82
4. Interpret	Answer in context	About 4.8 weeks until equal amounts

• Don't confuse: the variable represents the unknown you're solving for (weeks), not the money amounts themselves.

🔢 Step-by-step procedure

🔢 The four-step method

The excerpt outlines this process for equations with variables on both sides:

1. Use the Distributive Property to remove any parentheses.
2. Combine like terms on each side if needed.
3. Use the Addition Property of Equality to get the variable on one side and numerical values on the other.
4. Isolate the variable using the Multiplication Property of Equality (multiply or divide) to get the variable alone.

🔢 Why this order matters

• Parentheses must be removed first because you cannot accurately combine variable terms until all expressions are expanded.
• Variables must be gathered before isolating because you need all variable terms together to solve for the unknown.
• Example: Trying to isolate before gathering would leave variable terms on both sides, making it impossible to find a single value.

🧭 Overview

🧠 One-sentence thesis

Writing equations from real-world situations requires translating words and relationships into mathematical symbols, with key words like "exactly," "is," and "equivalent" signaling the equal sign.

📌 Key points (3–5)

• Translation process: Read every word in a situation and convert each into mathematical symbols to build an equation.
• Signal words for equality: Words like "exactly," "equivalent," "the same as," "identical," and "is" indicate where to place the equal sign.
• Three-step method: Choose a variable for the unknown, write an equation representing the situation, then solve by thinking through the relationship.
• Common confusion: Don't confuse expressions (no equal sign) with equations (must have an equal sign)—the presence of equality words determines which to write.

🔤 Signal Words for Equations

🔤 Words that mean "equals"

The excerpt identifies specific words that tell you to write an equation rather than just an expression:

These words can be used to symbolize the equal sign: Exactly, equivalent, the same as, identical, is

• These words are synonymous with the equal sign.
• When you see them, you know the situation requires an equation (not just an expression).
• Example: "Your family will spend exactly $25.00" → the word "exactly" signals you need an equals sign in your equation.

🆚 Equations vs expressions

• Expression: a mathematical phrase with no equal sign (covered in Chapter 2).
• Equation: a mathematical sentence with an equal sign showing two quantities are the same.
• The excerpt emphasizes: "Now we will practice writing equations" (building on earlier expression work).

🔢 The Three-Step Translation Method

📝 Step 1: Choose a variable

• Pick a letter to represent the unknown quantity.
• Use a letter that makes sense: b for burgers, n for "a number," m for money, x for a generic unknown.
• Example: "How many burgers can be purchased?" → Let b represent the number of burgers.

✍️ Step 2: Write the equation

• Translate each part of the word problem into mathematical symbols.
• Keep track of operations: "less than" means subtraction, "more than" means addition, "times" means multiplication.
• Combine all parts with the appropriate equal sign.
• Example: "Each burger costs $2.50" and "spend exactly $25.00" → 2.50b = 25.00

💭 Step 3: Think through the solution

• Ask yourself: "What number makes this equation true?"
• Example: "What number multiplied by 2.50 equals 25.00?" → The answer is 10.
• This step helps verify your equation makes logical sense.

📚 Translation Examples from the Excerpt

📚 Example: Burger purchase

Situation: Your family plans to only purchase burgers at $2.50 each with $25.00.

• Variable: b for burgers
• Equation: 2.50b = 25.00
• Solution: 10 burgers (because 2.50 times 10 equals 25.00)

📚 Example: "Less than" phrasing

Situation: "9 less than twice a number is 33"

• Variable: n for "a number"
• Translation breakdown:
  • "Twice a number" → 2n
  • "9 less than [that]" → 2n − 9
  • "is" → equals sign
• Equation: 2n − 9 = 33

Don't confuse: "9 less than twice a number" means you subtract 9 from 2n, not subtract 2n from 9.

📚 Example: "More than" phrasing

Situation: "Five more than four times a number is 21"

• Variable: x for "a number"
• Translation: "four times a number" → 4x; "five more than [that]" → 4x + 5
• Equation: 4x + 5 = 21

📚 Example: Fraction relationships

Situation: "$20.00 was one-quarter of the money spent on pizza"

• Variable: m for "the money"
• Translation: "one-quarter of the money" → (1/4)m; "was" → equals
• Equation: (1/4)m = 20.00

🏪 Real-World Application Scenarios

🏪 Service pricing equations

The excerpt provides several practice scenarios involving costs:

Situation	Given information	What to find
Peter's Lawn Mowing	$10 per job + $0.20 per square yard; earns $25 total	Equation relating job fee, area, and total
Ice-skating rink rental	$200 base + $4 per person; costs $324 total	Equation for number of people
Car rental	$55 per day + $0.45 per mile; costs $100 total	Equation for days and miles

• Each scenario has a fixed cost plus a variable cost.
• The equation structure: (fixed amount) + (rate × quantity) = total cost
• Example: For Peter's service, if y = square yards: 10 + 0.20y = 25

🏪 Other relationship types

The excerpt also includes:

• Simple addition: "Nadia gave Peter 4 more blocks than he already had. He already had 7 blocks."
• Interest problems: "Money invested at 5% annual interest earns $250."
• Budget constraints: "Hamburgers cost $0.49 each. You have $3 to spend."

Each requires identifying the unknown, the relationship, and the equality condition.

🧭 Overview

🧠 One-sentence thesis

Ratios and proportions provide a mathematical framework for comparing quantities and solving real-world problems involving relationships between measurements, enabling indirect calculations when direct measurement is impractical.

📌 Key points (3–5)

• What ratios and proportions are: A ratio compares two quantities as a fraction; a proportion states that two ratios are equal.
• Cross-products method: When two fractions form a proportion (a/b = c/d), then ad = bc, which provides a tool for solving unknown values.
• Unit consistency rule: When setting up proportions, keep the same units in corresponding positions in each fraction (e.g., miles/time = miles/time).
• Common confusion: Ratios vs rates—ratios compare quantities with the same units; rates compare quantities with different units (like miles per hour).
• Real applications: Proportions solve scale problems (maps, models), indirect measurements (distances that can't be measured directly), and conversion problems.

🔢 Understanding ratios and rates

🔢 What a ratio is

A ratio is a fraction comparing two things.

• Expresses the relationship between two quantities with the same units.
• Can be simplified like any fraction.
• Example: A room measuring 48 feet long by 36 feet wide has a length-to-width ratio of 48/36 = 4/3.

🚗 What a rate is

A rate is a fraction comparing two things with different units.

• Common rates include speed (65 mi/hour), price ($1.99/pound), and density ($3.79/yd²).
• The key difference: rates involve different units in numerator and denominator.
• Example: A student-to-teacher ratio shows how many students one teacher is responsible for.

✍️ Three ways to write ratios

Ratios can be expressed in multiple formats:

• Fraction form: a/b
• Colon notation: a:b (read as "a is to b")
• Words: "a is to b"

⚖️ Working with proportions

⚖️ What makes a proportion

A proportion is a statement that two fractions are equal: a/b = c/d.

• Not every pair of fractions forms a proportion—they must be truly equal.
• Example: Is 2/3 = 6/12 a proportion? Convert to common denominator: 2/3 = 8/12, but 6/12 ≠ 8/12, so this is NOT a proportion.

✖️ Cross-products rule

When a/b = c/d is a proportion, then ad = bc.

Why this works:

• Multiply both sides by b: a = bc/d
• Multiply both sides by d: ad = bc
• The cross-products are: ad (first numerator × second denominator) and bc (first denominator × second numerator)

Example: Solve a/9 = 7/6

• Cross-multiply: 6a = 7(9)
• Simplify: 6a = 63
• Solve: a = 10.5

🧩 Setting up proportions correctly

Critical rule: Keep units consistent in each fraction.

Correct setups:

• miles/time = miles/time
• miles/time ≠ time/miles (wrong—units are flipped)

Example scenario: A train covers 15 miles in 20 minutes. How far in 7 hours at the same rate?

• Set up: (miles/time) = (miles/time)
• Ensure time units match (convert 7 hours to 420 minutes)

📏 Scale and indirect measurement

📏 Map scales

The scale of a map describes the relationship between distances on a map and the corresponding distances on the earth's surface.

How scales work:

Scale notation	Meaning	Example
1:100,000	1 unit on map = 100,000 units in reality	1 cm on map = 1 km in real life
1:500	1 unit on map = 500 units in reality	Used for detailed drawings
1:1	Map same size as area shown	Impractical—map as large as reality!

🗺️ Solving scale problems

Example: On a 1:100,000 map, a distance measures 8.8 cm. What is the real distance?

Set up proportion:

• (distance on map)/(real distance) = 1/100,000
• 8.8 cm / x = 1/100,000
• Cross-multiply: 880,000 cm = x
• Convert: 8,800 m or 8.8 km

Don't confuse: The scale 1 cm = 1 km looks simple, but not all scales are this convenient—always refer to the actual scale ratio.

📐 Indirect measurement applications

Proportions enable measuring things that are difficult to measure directly:

• Height of tall trees
• Width of wide rivers
• Distance between cities across mountains
• Heights of lunar craters

The method links measurement with geometry using similar triangles and proportional relationships.

🎯 Practical problem-solving

🎯 Unit rates

A unit rate expresses a ratio per one unit of the denominator.

Examples from the excerpt:

• 54 hotdogs to 12 minutes → 4.5 hotdogs per minute
• 180 students to 6 teachers → 30 students per teacher
• 12 meters to 4 floors → 3 meters per floor

🧮 Real-world proportion problems

Restaurant revenue example: A restaurant serves 100 people and takes in $908. If it serves 250 people, what might revenue be?

Set up:

• people/revenue = people/revenue
• 100/$908 = 250/x
• Cross-multiply and solve for x

Ammonia compound example: Ammonia has a 1:3 ratio of nitrogen to hydrogen atoms. If a sample has 1,983 hydrogen atoms, how many nitrogen atoms?

• nitrogen/hydrogen = 1/3
• x/1,983 = 1/3
• Solve: x = 661 nitrogen atoms

🧭 Overview

🧠 One-sentence thesis

Scale and indirect measurement allow us to determine distances and sizes that are difficult to measure directly by using proportions, similar triangles, and scaled representations like maps.

📌 Key points (3–5)

• What indirect measurement solves: measuring things that are hard to reach directly (tall trees, wide rivers, distances between cities).
• How maps use scale: a map scale expresses the ratio between distances on the map and real distances on Earth's surface.
• Scale as a proportion: scale can be written as a fraction or ratio (e.g., 1:100,000 means 1 unit on map = 100,000 units in reality).
• Common confusion: not all scales are intuitive (like 1 cm = 1 km); you must always refer to the specific map scale to convert correctly.
• Why it matters: enables practical problem-solving for navigation, drawing, and measuring inaccessible objects.

📏 What is indirect measurement

📏 The core idea

Indirect measurement: making measurements of things that would be difficult to measure directly, using proportions and similar triangles.

• Instead of physically measuring something (like the height of a tall tree or width of a wide river), you use mathematical relationships.
• Links measurement with geometry and numbers.
• Examples of hard-to-measure things mentioned:
  • Height of a tall tree
  • Width of a wide river
  • Height of the moon's craters
  • Distance between cities separated by mountains

🔗 Connection to proportions

• The method relies on setting up proportions (equal ratios).
• You measure something accessible, then use the proportion to calculate the inaccessible measurement.

🗺️ Understanding map scales

🗺️ What a map scale represents

Map scale: describes the relationship between distances on a map and the corresponding distances on Earth's surface.

• A map is a two-dimensional, geometrically accurate representation of part of Earth's surface.
• The scale tells you how much real distance each unit on the map represents.
• Expressed as a fraction or ratio.

📝 How to write scales

Ratio notation	Word form	Fraction form	Meaning
1:20	one to twenty	1/20	1 unit on map = 20 units in reality
2:3	two to three	2/3	2 units on map = 3 units in reality
1:1000	one to one-thousand	1/1000	1 unit on map = 1000 units in reality

• Outside math books, ratios are often written with a colon (:).
• The excerpt notes: a 1:1 (one to one) map would be as large as the area it shows!

⚠️ Don't assume simplicity

• Not all map scales are as simple as "1 cm = 1 km."
• You must always refer to the specific map scale to convert between map measurements and real distances.
• Example: If the scale is 1:100,000, then 1 cm on the map represents 100,000 cm (or 1 km) in real life.

🧮 Solving scale problems with proportions

🧮 The proportion method

The general approach:

• Write the scale as a proportion: (distance on map) / (real distance) = (scale ratio)
• Substitute known values.
• Cross multiply to solve for the unknown.

🚶 Example: Navigation distance

Scenario: Anne measures 8.8 cm on a 1:100,000 scale map (where 1 cm on map = 1 km in reality). What is the real distance?

Solution steps:

1. Set up proportion: (distance on map) / (real distance) = 1 / 100,000
2. Substitute: 8.8 cm / x = 1 / 100,000
3. Cross multiply: 880,000 cm = x
4. Convert units: 880,000 cm = 8,800 m = 8.8 km
5. Answer: The real distance is 8.8 km.

Note: In this particular case, the 1 cm = 1 km scale means you could use intuition (reading in cm gives you km), but this won't work for all scales.

📐 Example: Scale drawing

Scenario: Oscar wants to draw the Titanic (883 feet long) at 1:500 scale. How long must his paper be in inches?

Solution steps:

1. Calculate scaled length: 883 / 500 = 1.766 feet
2. Convert to inches: 1.766 × 12 = 21.192 inches
3. Answer: The paper should be at least 22 inches long.

Key insight: The scale ratio tells you how much to divide the real measurement to get the drawing measurement.

🔢 Working with units in scale problems

🔢 Unit conversion is essential

• Map measurements and real distances often use different units.
• Common conversions mentioned:
  • 100 cm = 1 m
  • 1000 m = 1 km
  • 12 inches = 1 foot

🎯 Strategy for solving

1. Identify what you know (map distance or real distance).
2. Identify what you need to find.
3. Set up the proportion using the scale.
4. Cross multiply and solve.
5. Convert units as needed to get the answer in the requested form.

🧭 Overview

🧠 One-sentence thesis

Percent problems can be solved algebraically by translating word problems into equations using conversion techniques and the percent equation framework, which is essential for calculating changes, discounts, and proportional relationships.

📌 Key points (3–5)

• Converting between forms: decimals to percents (multiply by 100), percents to decimals (divide by 100), and fractions to percents (set up proportion with x/100).
• The percent equation: translates "is" as equals and "of" as multiplication to form algebraic equations.
• Percent of change formula: (final amount - original amount) / original amount × 100%, where positive values indicate increases and negative values indicate decreases.
• Common confusion: distinguishing between finding a percent of a number versus finding what percent one number is of another—the setup changes based on what is unknown.
• Real-world applications: sales tax, discounts, salary raises, commissions, and demographic calculations all use the same algebraic percent framework.

🔄 Converting between decimals, percents, and fractions

🔢 Decimal to percent

• Multiply the decimal by 100 and add the % sign.
• Example: 0.3786 becomes 0.3786 × 100 = 37.86%
• This shifts the decimal point two places to the right.

🔢 Percent to decimal

• Divide the percentage by 100 and drop the % sign.
• Example: 98.6% becomes 98.6 ÷ 100 = 0.986
• This shifts the decimal point two places to the left.

🔢 Fraction to percent

• Set up a proportion: the fraction equals x/100, where x is the unknown percent.
• Cross multiply and solve for x.
• Example: 3/5 = x/100 leads to 5x = 300, so x = 60, meaning 3/5 = 60%

📐 The percent equation

🔑 Translation rules

The percent equation: translates word problems by using "is" for the equals sign and "of" for multiplication.

• "What is 30% of 85?" becomes n = 30% × 85
• "50 is 15% of what number?" becomes 50 = 15% × w
• Always convert the percent to a decimal before calculating.

🧮 Three types of percent problems

Unknown element	Question format	Equation setup
The result	What is 30% of 85?	n = 0.30 × 85
The whole	50 is 15% of what?	50 = 0.15 × w
The percent	What percent of 7.2 is 45?	7.2 × n = 45

💡 Solving strategy

• Identify which quantity is unknown.
• Translate the words into an equation using the rules above.
• Convert percent to decimal (divide by 100).
• Solve the resulting algebraic equation.
• Example: "What is 30% of 85?" → n = 0.30 × 85 → n = 25.5

📊 Percent of change

📈 The percent change formula

Percent change = [(final amount - original amount) / original amount] × 100%

• A positive percent change represents an increase.
• A negative percent change represents a decrease.
• The original amount always goes in the denominator.

📈 Increase problems

• Example: A school of 500 students expects a 20% increase.
• Setup: 20% = [(x - 500) / 500] × 100%
• Divide both sides by 100%: 0.2 = (x - 500) / 500
• Multiply by 500: 100 = x - 500
• Add 500: x = 600 students
• Don't confuse: the 20% is applied to the original amount, not the final amount.

📉 Decrease problems (discounts)

• Example: A $150 mp3 player is 30% off.
• Setup: [(x - 150) / 150] × 100% = -30% (negative because it's a discount)
• Divide by 100%: (x - 150) / 150 = -0.3
• Multiply by 150: x - 150 = -45
• Add 150: x = $105
• The discount is subtracted from the original price.

🌍 Real-world applications

💼 Multi-step demographic calculations

The excerpt provides a detailed example using USDA employment data with 112,071 total employees:

Finding minority percentage:

• Total employees: 112,071
• Caucasian employees: 87,846
• Minority employees: 112,071 - 87,846 = 24,225
• Rate × 112,071 = 24,225
• Rate ≈ 0.216 or 21.6%

Finding specific group percentage:

• African-American employees: 11,754
• Rate × 112,071 = 11,754
• Rate ≈ 0.105 or 10.5%

Finding remaining percentage:

• Other minorities: 24,225 - 11,754 - 6,899 = 5,572
• Rate × 112,071 = 5,572
• Rate ≈ 0.05 or 5%

🛍️ Sales, tax, and commission scenarios

• Sales tax: Add the tax percentage to the original price (e.g., $35 item with 7.75% tax).
• Commission: Multiply the sale price by the commission rate (e.g., 7.5% of $215,000).
• Discounts: Subtract the discount percentage from the original price, treating it as a negative percent change.
• Combined discounts: Apply employee discounts before subtracting coupons, as order matters.

🔍 Working backward from new values

• If you know the new salary after a raise, work backward to find the original.
• Example: $45,000 is the salary after a 5% raise.
• Setup: 1.05 × (original salary) = $45,000
• Original salary = $45,000 / 1.05

🧭 Overview

🧠 One-sentence thesis

Literal equations are formulas with multiple variables that can be rearranged to solve for any variable of interest by using the same algebraic techniques used to solve single-variable equations.

📌 Key points (3–5)

• What literal equations are: equations with several variables (more than one), such as formulas for area, perimeter, distance, and interest.
• Why rearranging matters: sometimes we know different variables than the formula solves for, so we need to isolate a different variable.
• How to rearrange: use the same techniques as solving regular equations—add, subtract, multiply, or divide both sides to isolate the target variable.
• Common confusion: when the variable you want is in a denominator or mixed with multiple terms, you must first clear fractions or move other terms before isolating.
• Practical use: rearranging formulas lets you solve real-world problems when you know different information than the standard formula expects.

📐 What literal equations are

📐 Definition and common examples

A literal equation is an equation with several variables (more than one).

Literal equations are formulas we use regularly. The excerpt provides these common examples:

Formula	Variables	What it calculates
A = LW	Area, Length, Width	Area of a rectangle
P = 2L + 2W	Perimeter, Length, Width	Perimeter of a rectangle
A = π r²	Area, radius	Area of a circle
F = 1.8C + 32	Fahrenheit, Celsius	Temperature conversion
D = RT	Distance, Rate, Time	Distance traveled
A = P + Prt	Accrued value, Principal, rate, time	Simple interest

📐 Why they're called "literal"

• The term "literal" refers to having multiple letter variables, not just one unknown.
• Each variable represents a different quantity in the relationship.

🔄 Rearranging formulas

🔄 When and why to rearrange

The excerpt explains that sometimes we need to rearrange a formula to solve for a different variable:

• We may know the Area and Length of a rectangle and want to find the Width.
• We may know the time and distance traveled and want to find the average speed (rate).
• The goal is to isolate the variable of interest on one side of the equal sign, with all other terms on the opposite side.

🔄 Key principle

• Use the same techniques that work for solving single-variable equations.
• The algebraic operations (add, subtract, multiply, divide) apply the same way.
• Whatever you do to one side, do to the other side.

🧮 Simple rearrangement examples

🧮 Finding width from area (Example 1)

Problem: Find the width of a rectangle given the area and length.

Starting formula: A = lw (Area = Length times Width)

Steps to solve for w:

1. Start: A = lw
2. Divide both sides by length: A/l = lw/l
3. Result: A/l = w

Why this works: Dividing both sides by l cancels the l on the right, leaving w isolated.

🧮 Finding voltage from power (Example 4)

Problem: Find voltage given current and power.

Starting formula: I = P/E (Current = Power/Voltage)

Challenge: The variable E (voltage) is in the denominator.

Steps to solve for E:

1. Start: I = P/E
2. Multiply both sides by E: (E/1) × I = (P/E) × (E/1)
3. Simplify: EI = P (now E is in the numerator)
4. Divide both sides by I: EI/I = P/I
5. Result: E = P/I

Don't confuse: When the target variable is in a denominator, you must first multiply to move it to a numerator before you can isolate it.

🔧 Multi-step rearrangement examples

🔧 Finding width from perimeter (Example 2)

Problem: Find the width of a rectangle when given perimeter and length.

Starting formula: P = 2l + 2w (Perimeter = twice the length plus twice the width)

Challenge: The formula has two terms on the right side.

Steps to solve for w:

1. Start: P = 2l + 2w
2. Subtract 2l from both sides: P - 2l = 2l + 2w - 2l
3. Simplify: P - 2l = 2w (now only one term on the right)
4. Divide both sides by 2: (P - 2l)/2 = 2w/2
5. Result: (P - 2l)/2 = w

Important: Move extra terms to the other side first, then divide to isolate the variable.

🔧 Finding one side of a triangle (Example 3)

Problem: Find the length of one side of a triangle given the perimeter and the other two sides.

Starting formula: P = a + b + c (Perimeter = sum of the three sides)

The excerpt notes "this one seems to give people problems," so let's solve for c:

Steps:

1. Start: P = a + b + c
2. Subtract a from both sides: P - a = a + b + c - a
3. Simplify: P - a = b + c
4. Subtract b from both sides: P - a - b = b + c - b
5. Result: P - a - b = c

Why it's tricky: You must subtract each unwanted variable one at a time; you can't skip steps.

🛠️ Real-world application

🛠️ Building a planter (Example 5)

Scenario: Dave is building a planter 2 feet wide and 4 feet long. He wants it to hold 20 cubic feet of soil. How high should it be?

Formula: V = lwh (Volume = length times width times height)

Solution process:

1. Rearrange to solve for h: V/lw = h
2. Plug in known values: 20/(2 × 4) = h
3. Calculate: 20/8 = 2.5
4. Answer: The planter should be 2.5 feet high.

Why rearranging first helps: Dave solved for h algebraically, then substituted numbers. This approach works for any similar problem, not just this one case.

🛠️ General strategy

• Identify which variable you need to find.
• Rearrange the formula to isolate that variable.
• Substitute the known values.
• Calculate the answer.

🧭 Overview

🧠 One-sentence thesis

Inequalities express mathematical relationships where values are not equal but instead greater than, less than, or within a range, and they can be solved using properties similar to equations while representing solutions in multiple ways.

📌 Key points (3–5)

• What inequalities are: mathematical sentences connecting expressions with inequality signs (>, ≥, ≤, <, ≠) instead of equal signs.
• Multiple solutions: unlike equations, inequalities typically have more than one solution, forming a solution set.
• Three ways to express solutions: inequality notation (e.g., x ≤ 4), interval notation using brackets, and graphed number lines.
• Common confusion: checking solutions requires choosing a value within the solution set, not the endpoint, because you need to test the direction of the inequality.
• Solving with addition/subtraction: you can add or subtract the same value from both sides to isolate the variable, just like with equations.

🔤 Understanding inequality symbols

🔤 The five main inequality signs

The excerpt lists the most common inequality signs:

Symbol	Meaning
>	greater than
≥	greater than or equal to
≤	less than or equal to
<	less than
≠	not equal to

• The line underneath (≥ or ≤) stands for "or equal to."
• These symbols replace the equal sign when exact equality is not required.

📝 What an algebraic inequality is

Algebraic inequality: a mathematical sentence connecting an expression to a value, variable, or another expression with an inequality sign.

Examples from the excerpt:

• 3x < 5
• (3x divided by 4) ≥ (x squared minus 3)
• 4 - x ≤ 2x

🌍 Real-world translation

Example: "Avocados cost $1.59 per pound. How many pounds can be purchased for less than $7.00?"

• Choose a variable (a for pounds of avocados).
• Write the inequality: 1.59(a) < 7
• The inequality captures "less than" rather than "exactly equal to."

✅ Checking and graphing solutions

✅ Why checking is different from equations

• Inequalities typically have more than one solution (a solution set).
• You cannot just substitute one value; you must verify the entire range.
• Key rule: choose a number that occurs within the solution set, not the endpoint.

Example from the excerpt: Check if m ≤ 10 is a solution to 4m + 30 ≤ 70.

• Choose a value less than 10, say 4.
• Substitute: 4(4) + 30 = 16 + 30 = 46 ≤ 70 ✓
• The excerpt explains: "Endpoints are not chosen when checking an inequality because the direction of the inequality needs to be tested."

📊 Graphing on a number line

• Filled circle: the value is included (≤ or ≥).
• Unfilled circle: the value is not included (< or >).
• Arrow direction: shows which side of the number line contains solutions.

Example: t > 3 means all real numbers greater than 3, shown with an unfilled circle at 3 and an arrow pointing right.

Example: The graph with a filled circle at 4 and an arrow pointing left represents x ≤ 4.

📐 Three ways to express solutions

📐 Inequality notation

• The answer is expressed as an algebraic inequality.
• Example: d ≤ (1 divided by 2)

📐 Interval notation

Uses brackets to denote the range of values:

• Square or closed brackets [ ]: the number is included in the solution.
• Round or open brackets ( ): the number is not included in the solution.
• Uses infinity (∞) and negative infinity (-∞).

Examples from the excerpt:

• (8, 24): all numbers between 8 and 24 but does not include 8 and 24.
• [3, 12): all numbers between 3 and 12, including 3 but not including 12.
• For d ≤ (1 divided by 2): (-∞, (1 divided by 2)]

📐 Graphed sentence on a number line

• Visual representation showing the range and whether endpoints are included.
• Combines the circle notation (filled/unfilled) with directional arrows.

🔧 Solving inequalities with addition and subtraction

🔧 Addition Property of Inequality

Addition Property of Inequality: For all real numbers a, b, and c:

• If x < a, then x + b < a + b
• If x < a, then x - c < a - c

• These properties also work for ≤ and ≥.
• Subtraction can be thought of as "add the opposite," so these properties cover subtraction situations.
• Goal: isolate the variable by canceling operations using inverses.

🧮 Solving example with subtraction

Example: Solve x - 3 < 10

• To isolate x, cancel "subtract 3" using addition.
• Add 3 to both sides: x - 3 + 3 < 10 + 3
• Simplify: x < 13
• Check: Choose a number less than 13 (e.g., 0): 0 - 3 = -3 < 10 ✓

🧮 Solving example with addition

Example: Solve x + 4 > 13

• Subtract 4 from both sides: x + 4 - 4 > 13 - 4
• Simplify: x > 9

Don't confuse: The process is similar to solving equations, but the solution is a range of values, not a single number.

🌎 Writing real-life inequalities

🌎 Translating statements to inequalities

Real-world situations often require inequalities rather than exact equations.

Example: "You must maintain a balance of at least $2,500 in your checking account to avoid a finance charge."

• Key phrase: "at least" means $2,500 or more.
• Choose variable m for money in account.
• Write: m ≥ 2500
• Graph with a filled circle at 2500 and arrow pointing right.

Example: "The speed limit is 65 miles per hour."

• To avoid a ticket, drive 65 or less.
• Choose variable s for speed.
• Write: s ≤ 65
• Graph with a filled circle at 65 and arrow pointing left.
• Note: The excerpt mentions that "in theory, you cannot drive a negative number of miles per hour," acknowledging practical constraints.

🌎 Identifying key phrases

• "At least" → ≥
• "More than" → >
• "Less than" → <
• "No more than" → ≤

Example: "Charlie needs more than $1,800 to purchase a car" translates to a variable > 1800.

🧭 Overview

🧠 One-sentence thesis

Solving inequalities by multiplication or division follows the same logic as equations except that multiplying or dividing both sides by a negative number reverses the inequality sign.

📌 Key points (3–5)

• Multiplication and division properties: When multiplying or dividing both sides by a positive number, the inequality direction stays the same.
• The negative-number rule: When multiplying or dividing both sides by a negative number, you must flip the inequality sign (< becomes >, ≤ becomes ≥, etc.).
• Why the sign flips: Multiplying by -1 produces the opposite of both values, so the relationship reverses (e.g., if x > 4, then -x < -4).
• Common confusion: Students often forget to reverse the sign when dividing or multiplying by a negative; remember that only negative multipliers/divisors trigger the flip.
• Solution formats: Solutions can be expressed as an inequality, set notation, interval notation, or a number line graph.

⚖️ Core properties for positive numbers

➕ Multiplication Property of Inequality

For all real positive numbers a, b, and c: If x < a, then x times c < a times c. If x > a, then x times c > a times c.

• This property applies only when c is positive.
• The inequality direction remains unchanged.
• Example: If 2x ≥ 12, divide both sides by 2 (positive) to get x ≥ 6; the ≥ sign stays the same.

➗ Division Property of Inequality

For all real positive numbers a, b, and c: If x < a, then x ÷ c < a ÷ c. If x > a, then x ÷ c > a ÷ c.

• Again, this applies only when c is positive.
• Dividing by a positive number preserves the inequality direction.
• Example: If y/5 ≤ 3, multiply both sides by 5 to get y ≤ 15; the ≤ sign does not change.

🔄 The negative-number rule

🔄 Why the sign reverses

• When you multiply any value by -1, you get its opposite: 6 times -1 = -6.
• Multiplying both sides of an inequality by -1 reverses everything, including the inequality sign.
• Example: If x > 4, then multiplying both sides by -1 gives -x < -4 (the > flips to <).
• The excerpt says: "When multiplying by a negative, you are doing the opposite of everything in the sentence, including the verb."

⚠️ Multiplication/Division Rule of Inequality

For any real number a and any negative number c: If x < a, then x times c > a times c. If x < a, then x/c > a/c.

• These rules also hold for ≤ and ≥.
• Don't confuse: Only when c is negative do you reverse the sign; positive c does not trigger a flip.

🧮 Example with a negative divisor

• Problem: -3r < 9
• Divide both sides by -3 (negative), so flip the sign: r > -3.
• The < becomes > because you divided by a negative number.

🧮 Example with a positive divisor

• Problem: 12p < -30
• Divide both sides by 12 (positive), so do not flip the sign: p < -5/2.
• Because 12 is not negative, the inequality direction stays the same.

📝 Expressing solutions

📝 Four solution formats

The excerpt shows that solutions can be written in four ways:

Format	Example (for x ≥ 6)
Inequality notation	x ≥ 6
Set notation	{x | x ≥ 6}
Interval notation	[6, ∞)
Number line graph	A ray starting at 6, extending right, with a closed dot at 6

• All four formats convey the same information.
• Choose the format requested by the problem or instructor.

📝 Interval notation details

• For y ≤ 15, interval notation is (-∞, 15].
• For p < -5/2, interval notation is (-∞, -5/2).
• Use square brackets [ ] for ≤ or ≥ (inclusive), parentheses ( ) for < or > (exclusive).

🧭 Overview

🧠 One-sentence thesis

Multi-step inequalities are solved using the same procedure as multi-step equations, except that multiplying or dividing by a negative number requires reversing the inequality sign.

📌 Key points (3–5)

• Core process: Multi-step inequalities follow the same solving steps as multi-step equations (distributive property, combining like terms, isolating the variable).
• Critical difference: When multiplying or dividing both sides by a negative number, you must reverse the inequality sign (< becomes >, ≤ becomes ≥).
• Solution types: Inequalities can have infinitely many solutions, no solutions, or a finite set of solutions depending on the context.
• Common confusion: Forgetting to flip the inequality sign when dividing/multiplying by a negative—this is the only exception to treating inequalities like equations.
• Real-world application: Word problems often involve constraints like "at least" (≥) or "at most" (≤) that translate into inequalities.

🔧 The solving procedure

📋 Five-step checklist

The excerpt provides a systematic procedure:

1. Remove parentheses using the Distributive Property
2. Simplify each side by combining like terms
3. Isolate the variable term (the "ax" term) using the Addition Property of Inequality
4. Isolate the variable using the Multiplication/Division Property—reverse the inequality sign if multiplying or dividing by a negative
5. Check your solution by substituting a test value

⚠️ The critical rule

When multiplying or dividing by a negative number, reverse the inequality sign.

• This is the only major difference from solving equations.
• Example from the excerpt: When solving negative 4x < negative 15, dividing both sides by negative 4 changes < to >, giving x > 15/4.
• Don't confuse: All other steps (adding, subtracting, distributive property) work exactly like equations—no sign reversal needed.

🧮 Worked examples breakdown

🔢 Basic multi-step inequality

Example: 6x − 5 < 10

• No parentheses, no like terms to combine
• Add 5 to both sides: 6x < 15
• Divide by 6 (positive, so no sign flip): x < 2.5
• Check with a value less than 2.5, such as 0: 6(0) − 5 = −5, which is indeed < 10

➖ Dividing by a negative

Example: −9x < −5x − 15

• Add 5x to both sides: −4x < −15
• Divide by −4: must reverse the sign from < to >
• Result: x > 15/4, which equals x > 3.75
• Check with a value greater than 3.75, such as 10

🎯 Complex inequality with parentheses

Example: 4x − 2(3x − 9) ≤ −4(2x − 9)

1. Distribute: 4x − 6x + 18 ≤ −8x + 36
2. Combine like terms: −2x + 18 ≤ −8x + 36
3. Add 8x to both sides: 6x + 18 ≤ 36
4. Subtract 18: 6x ≤ 18
5. Divide by 6 (positive): x ≤ 3

🔍 Types of solutions

♾️ Infinitely many solutions

• Most inequalities solved in the excerpt have infinite solutions.
• Example: x ≤ 3 means any number 3 or smaller works.

∅ No solutions

Example from the excerpt: x − 5 > x + 6

• Subtract x from both sides: −5 > 6
• This statement is always false (negative five is never greater than six).
• Therefore, the inequality has no solutions.

📏 Finite solution set

Example: Speed limit problem where s ≤ 65

• Speed cannot be negative, so the context restricts solutions.
• Solution set in interval notation: [0, 65]
• This is a finite range, not all numbers less than or equal to 65.

🌍 Real-world applications

📰 Subscription sales problem

Problem: Leon must sell at least 120 subscriptions; he already sold 85. How many more does he need?

• "At least 120" translates to ≥ 120
• Inequality: n + 85 ≥ 120
• Solve: n ≥ 35
• Leon must sell 35 or more subscriptions in the last week.

📐 Rectangle perimeter problem

Problem: Rectangle width is 12 inches; perimeter must be at least 180 inches. Find the length.

• Perimeter formula: 12 + 12 + x + x ≥ 180
• Simplify: 24 + 2x ≥ 180
• Solve: 2x ≥ 156, so x ≥ 78
• The length must be 78 inches or larger.

🔑 Translation keywords

Phrase	Inequality symbol
At least	≥
At most	≤
Greater than	>
Less than	<

🧭 Overview

🧠 One-sentence thesis

The coordinate plane enables us to visually represent and locate points using two perpendicular number lines, forming the foundation for graphing equations and analyzing data relationships.

📌 Key points (3–5)

• What the coordinate plane is: two perpendicular number lines (x-axis and y-axis) that meet at the origin and divide space into four quadrants.
• How to read coordinates: ordered pairs (x, y) describe a point's position by stating horizontal distance first, then vertical distance from the origin.
• How to navigate: positive x means right, negative x means left; positive y means up, negative y means down from the origin.
• Common confusion: scale and range—axes don't always count by ones, and the two axes can have different scales; tick marks must be equally spaced on each axis.
• Why it matters: coordinate planes allow precise plotting of data points and form the basis for graphing equations and visualizing mathematical relationships.

📐 Structure of the coordinate plane

📐 The two axes

The x-axis is the horizontal line; the y-axis is the vertical line. Together they are called the axes.

• The coordinate plane is essentially two number lines meeting at right angles.
• The point where they cross is called the origin.
• Each axis extends in both positive and negative directions.

🔲 The four quadrants

Quadrants: the four regions created when the axes split the coordinate plane.

• Numbered I, II, III, IV moving counter-clockwise from the upper right.
• Quadrant I: upper right (positive x, positive y)
• Quadrant II: upper left (negative x, positive y)
• Quadrant III: lower left (negative x, negative y)
• Quadrant IV: lower right (positive x, negative y)
• Points on the axes themselves are not in any quadrant.

🎯 Reading and writing coordinates

🎯 What ordered pairs mean

Ordered pair: two numbers written in parentheses that describe a point's location, with the x-coordinate first and y-coordinate second.

• Format: (x, y)
• The x-coordinate (also called abscissa) tells horizontal distance from the origin.
• The y-coordinate (also called ordinate) tells vertical distance from the origin.
• Example: (3, 7) means 3 units right and 7 units up from the origin.

🧭 How to find coordinates from a point

Think of standing at the origin and moving to align with the point:

1. Horizontal movement first: count units right (positive) or left (negative) to get the x-coordinate.
2. Vertical movement second: count units up (positive) or down (negative) to get the y-coordinate.

Example: To reach point Q at (3, -2), move 3 units right (positive x-direction), then 2 units down (negative y-direction).

⚠️ Direction signs matter

Direction	Sign	Coordinate
Right	Positive	+x
Left	Negative	-x
Up	Positive	+y
Down	Negative	-y

Don't confuse: The order matters—(3, -2) is not the same as (-2, 3). Always write x first, then y.

📍 Plotting points on the plane

📍 The plotting process

To plot a point like A(2, 7):

1. Start at the origin (0, 0).
2. Move horizontally according to the x-coordinate (2 units right).
3. From that position, move vertically according to the y-coordinate (7 units up).
4. Mark the point where you end up.

🔢 Working with different number types

Coordinates can be:

• Whole numbers: (3, -3)
• Decimals: (2.5, 0.5)
• Fractions: (1/2, -3/4)
• Irrational numbers: (π, 1.2)
• Zero: (0, 2) lies on an axis; (0, 0) is the origin

Example: Point E(0, 2) is 2 units up from the origin and sits right on the y-axis, between Quadrants I and II.

📊 Scale and range considerations

📊 Choosing appropriate scale

Scale: the increment by which tick marks on an axis increase; tick marks must be equally spaced on each axis.

Key principles:

• Tick marks don't have to count by ones—they can use increments of 2, 5, 0.5, or any consistent value.
• Choose increments that maximize clarity for your data.
• Each axis can have a different scale, but each individual axis must be consistent.

📏 Setting the range

Range: the portion of the coordinate plane you choose to display.

Guidelines from the excerpt:

• Start at or just below the lowest value in your data.
• End at or just above the highest value.
• If you don't start at zero, use "//" between the origin and the first tick mark to show the break.
• You don't always need to show all four quadrants—if all data is positive, showing only Quadrant I may be clearer.

Example: For points where x-values range from 0 to 3.14 and y-values from 0 to 1.75, showing 0 ≤ x ≤ 3.5 and 0 ≤ y ≤ 2 is more useful than showing all four quadrants.

⚠️ Common scale mistakes to avoid

• Unequal spacing: Tick marks on the same axis must represent equal increments (like a number line).
• Ignoring data range: Showing too much or too little of the plane can make points hard to read.
• Forgetting labels: Always label what each axis represents and what units are being used.

Don't confuse: Distances that look the same on both axes may represent different actual values if the axes use different scales.

🧭 Overview

🧠 One-sentence thesis

A good graph must have properly labeled axes with units, consistent and appropriate scales, and clear presentation that maximizes readability and accurate interpretation of the data.

📌 Key points (3–5)

• Labeling is critical: both axes must show variable names and units; without units, the graph becomes uninterpretable.
• Scale consistency matters: tick marks should be equally spaced, start at or near the lowest value, and use sensible increments for the data.
• Common confusion: axes don't need identical scales—horizontal and vertical can use different increments as long as each is consistent within itself.
• Space usage: good graphs make efficient use of available space and choose numbers that make sense for the dataset.
• Reading graphs: interpreting graphs involves finding values on one axis, tracing to the function line, then reading the corresponding value on the other axis.

📏 Essential elements of good graphs

📏 Axis labeling requirements

The horizontal axis should be properly labeled with the name and units of the input variable; the vertical axis should be properly labeled with the name and units of the output variable.

• Variable name: what is being measured (e.g., Weight, Time, Temperature).
• Units: the measurement unit (e.g., pounds, seconds, degrees Fahrenheit).
• Without units, readers cannot tell whether "Time" means minutes, seconds, or hours.
• Example: A weight axis labeled only "Weight" is incomplete; it should specify "Weight (lbs)" or "Weight (kg)."

📐 Scale requirements

The excerpt lists specific scale guidelines:

• Start at or just below the lowest data value.
• End at or just above the highest data value.
• Adjacent tick marks must be equal distance apart (consistency).
• Use numbers that make sense for the dataset.
• If the scale doesn't begin at zero, use "//" between the origin and the first tick mark.

Don't confuse: The two axes don't need the same scale—one might count by 1s while the other counts by 5s—but each axis must be internally consistent.

🎯 Space optimization

• Choose scales that maximize clarity and make good use of available space.
• Poor space usage makes graphs harder to read and interpret.
• Example: If data ranges from 60 to 90, starting the axis at 0 wastes space; starting near 60 is more efficient.

❌ Common graphing mistakes

❌ Missing information

The excerpt provides a "bad graph" example showing what NOT to do:

• Missing units on horizontal axis: "Time" without specifying minutes, seconds, etc.
• Missing variable name on vertical axis: shows "Feet" but doesn't say what is measured in feet.
• These omissions make the graph impossible to interpret correctly.

❌ Inconsistent or poor scales

Problems identified in the bad graph example:

• Horizontal scale is not consistent (unequal spacing between tick marks).
• Vertical scale does not start at zero (without proper notation).
• Very poor use of available space.

📖 Reading and interpreting graphs

📖 Basic reading technique

The excerpt describes a systematic process:

1. Find the known value on the appropriate axis (horizontal or vertical).
2. Draw a line from that point until it meets the function/data line.
3. Draw a perpendicular line to the other axis.
4. Read the corresponding value where this line intersects the axis.

Example: To find the price with tax for a $6.00 item, locate x = 6 on the horizontal axis, trace up to the function line, then trace horizontally to read approximately $6.75 on the vertical axis.

📖 Conversion graphs

The excerpt demonstrates reading conversion graphs (Fahrenheit to Celsius, kilograms to pounds):

• These show relationships between two linked quantities.
• You can convert in either direction by starting from either axis.
• Example: To convert 30° Celsius to Fahrenheit, find y = 30 on the Celsius axis, trace horizontally to the function, then trace down to read approximately 85° Fahrenheit.

📖 Real-world applications

The excerpt mentions practical uses:

• Stock price fluctuations over time.
• Blog readership trends.
• Temperature predictions from weather reports.
• Carbon dioxide concentration changes over years.

Key insight: Analyzing graphs is "a part of life"—a fundamental skill for interpreting data in many contexts.

🧭 Overview

🧠 One-sentence thesis

Graphing linear equations by plotting points from a table of values reveals straight-line relationships that can be used to solve real-world problems and predict unknown values.

📌 Key points (3–5)

• How to graph a linear equation: create a table of values by substituting x-values into the equation, plot the resulting (x, y) points, and connect them with a straight line.
• What the graph reveals: the y-intercept (where the line crosses the y-axis) corresponds to the constant term in the equation, and the coefficient of x determines how steeply the line rises or falls.
• Special cases—horizontal and vertical lines: equations of the form y = constant produce horizontal lines; equations of the form x = constant produce vertical lines (which are relations, not functions).
• Common confusion: horizontal vs. vertical—y = constant means y stays the same for all x (horizontal); x = constant means x stays the same for all y (vertical).
• Why it matters: graphs allow you to estimate values between data points and solve practical problems by reading coordinates directly from the line.

📊 Creating and interpreting linear graphs

📝 Making a table of values

• Choose several x-values (typically 0, 1, 2, 3, 4, or values that fit the problem context).
• Substitute each x-value into the equation to calculate the corresponding y-value.
• Record the pairs in a table.
• Tip: choose a scale that accommodates the range of values you need to analyze (e.g., if the problem asks about x = 7, make sure your graph extends to at least x = 7).

📍 Plotting points and drawing the line

• Plot each (x, y) pair from your table on a coordinate plane.
• Use a ruler to draw a straight line through all the points.
• The line extends beyond the plotted points, allowing you to estimate values not in your original table.

Example: For the taxi equation y = 0.8x + 3, a table might include (0, 3), (1, 3.8), (2, 4.6), (3, 5.4), (4, 6.2). Plotting these points and connecting them produces a straight line.

🔍 Reading values from the graph

• To find an output (y) for a given input (x): locate x on the horizontal axis, draw a vertical line up to the graph, then draw a horizontal line across to the y-axis and read the value.
• To find an input (x) for a given output (y): locate y on the vertical axis, draw a horizontal line across to the graph, then draw a vertical line down to the x-axis and read the value.

Example: To find the cost of a seven-mile taxi ride from y = 0.8x + 3, locate x = 7, trace up to the line, then across to the y-axis; the value is approximately 8.5, meaning $8.50.

🔑 Key features of linear graphs

📐 The straight-line property

A linear equation produces a graph that is a straight line.

• "Linear" means the relationship between x and y is constant—every time x increases by a fixed amount, y changes by a fixed amount.
• This is why you only need a few points to draw the entire line.

🎯 The y-intercept

• The y-intercept is where the line crosses the y-axis (where x = 0).
• In the equation, the y-intercept is the constant term added or subtracted at the end.
• Example: In y = 0.8x + 3, the y-intercept is 3. In y = -85x + 500, the y-intercept is 500.
• What it means: the y-intercept often represents a starting value or fixed cost in real-world problems (e.g., the base fee for a taxi, the initial debt of a business).

📈 The slope (rate of change)

• The coefficient of x tells you how much y changes for each one-unit increase in x.
• Example: In y = 0.8x + 3, every time x increases by 1, y increases by 0.8. In y = -85x + 500, every time x increases by 1, y decreases by 85.
• What it means: the slope represents the rate of change (e.g., cost per mile, debt paid per year).
• If the coefficient is negative, the line slopes downward (y decreases as x increases).

➖ Horizontal and vertical lines

➖ Horizontal lines: y = constant

An equation of the form y = constant produces a horizontal line that crosses the y-axis at the value of the constant.

• The y-value is the same for every x-value.
• Example: y = 7.5 means no matter what x is, y is always 7.5. The graph is a horizontal line at y = 7.5.
• Real-world meaning: a flat rate that doesn't depend on the input variable (e.g., a taxi charging $7.50 for any distance within city limits).

| Vertical lines: x = constant

An equation of the form x = constant produces a vertical line that crosses the x-axis at the value of the constant.

• The x-value is the same for every y-value.
• Example: x = 4 means no matter what y is, x is always 4. The graph is a vertical line at x = 4.
• Important distinction: vertical lines are relations, not functions, because one x-value corresponds to infinitely many y-values (violating the definition of a function).

🔄 Don't confuse horizontal and vertical

Type	Equation form	Direction	Example	Intercept
Horizontal	y = constant	Parallel to x-axis	y = 4	Crosses y-axis at 4
Vertical	x = constant	Parallel to y-axis	x = 4	Crosses x-axis at 4

🎯 The axes themselves

• The x-axis is the horizontal line where y = 0.
• The y-axis is the vertical line where x = 0.
• They intersect at the origin (0, 0).

🧮 Worked examples

🚖 Example: Taxi cost

• Equation: y = 0.8x + 3 (where x is miles traveled, y is cost in dollars).
• Table: (0, 3), (1, 3.8), (2, 4.6), (3, 5.4), (4, 6.2).
• Graph: a straight line starting at (0, 3) and rising steadily.
• To find the cost of a 7-mile ride: trace from x = 7 up to the line, then across to y ≈ 8.5, so approximately $8.50 (exactly $8.60 by calculation).

💼 Example: Business debt

• Equation: y = -85x + 500 (where x is years in business, y is debt in thousands of dollars).
• Table: (0, 500), (1, 415), (2, 330), (3, 245), (4, 160).
• Graph: a straight line starting at (0, 500) and sloping downward.
• To find when the debt is paid off: locate where the line crosses y = 0 (the x-axis); this occurs at approximately x = 6, meaning the debt will be paid off in 6 years.

🚕 Example: Flat-rate taxi

• Equation: y = 7.5 (cost is $7.50 regardless of distance).
• Graph: a horizontal line at y = 7.5.
• The x-variable (distance) does not appear in the equation because it has no effect on the cost.

⚠️ Important reminders

📏 Choosing an appropriate scale

• Make sure your graph extends far enough to answer the question.
• If you need to find a value at x = 7, don't stop your x-axis at x = 4.
• Leave a little extra space above and below the range of interest for clarity.

🔢 Discrete vs. continuous

• Although we draw the function as a continuous line for easier interpretation, the actual situation may be discrete (only certain values make sense).
• Example: you can't buy 2.3 taxi rides, but the line helps you estimate costs for any distance.

🧩 Reading the graph carefully

• When estimating from a graph, read values as accurately as possible (e.g., "halfway between 8 and 9" suggests 8.5).
• For exact answers, substitute the value into the equation and calculate.

🧭 Overview

🧠 One-sentence thesis

Intercepts—the points where a line crosses the x-axis and y-axis—provide a quick two-point method for graphing linear equations, and the cover-up method makes finding intercepts in standard form especially efficient.

📌 Key points (3–5)

• What intercepts are: the y-intercept occurs where the graph crosses the y-axis (x = 0), and the x-intercept occurs where the graph crosses the x-axis (y = 0).
• Why two points are enough: infinitely many lines pass through one point, but only one line passes through two points, so knowing both intercepts lets you graph the line.
• How to find intercepts algebraically: substitute x = 0 to find the y-intercept and y = 0 to find the x-intercept, then solve.
• The cover-up shortcut: for equations in standard form (ax + by = c), "cover up" the x-term to find the y-intercept and cover up the y-term to find the x-intercept.
• Common confusion: not all lines have both intercepts—horizontal lines never cross the x-axis and vertical lines never cross the y-axis.

📍 Understanding intercepts

📍 What intercepts are

y-intercept: the point where the graph crosses the y-axis; at this point, x = 0.

x-intercept: the point where the graph crosses the x-axis; at this point, y = 0.

• The excerpt emphasizes that intercepts are coordinates: the y-intercept has the form (0, y-value) and the x-intercept has the form (x-value, 0).
• Example: if a line crosses the y-axis at y = 8, the y-intercept is (0, 8); if it crosses the x-axis at x = 6, the x-intercept is (6, 0).

🔢 Why two points are sufficient

• The excerpt explains that infinitely many lines pass through a single point, so one point is not enough information to graph a line.
• But only one line passes through two specific points.
• How to use intercepts: plot both intercept points, then use a ruler to draw the line that passes through both.

⚠️ Special cases without both intercepts

• Horizontal lines never cross the x-axis (they run parallel to it).
• Vertical lines never cross the y-axis (they run parallel to it).
• The excerpt notes that "not all lines will have both an x- and a y-intercept, but most do."

🔍 Finding intercepts by substitution

🔍 The substitution method

• To find the y-intercept: substitute x = 0 into the equation and solve for y.
• To find the x-intercept: substitute y = 0 into the equation and solve for x.

🧮 Step-by-step examples

Example: y = 13 − x

• y-intercept: substitute x = 0 → y = 13 − 0 = 13 → (0, 13).
• x-intercept: substitute y = 0 → 0 = 13 − x → x = 13 → (13, 0).
• Then plot these two points and join them with a line.

Example: y = 2x + 3

• y-intercept: x = 0 → y = 2(0) + 3 = 3 → (0, 3).
• x-intercept: y = 0 → 0 = 2x + 3 → subtract 3 → −3 = 2x → divide by 2 → x = −1.5 → (−1.5, 0).

Example: 4x − 2y = 8

• y-intercept: x = 0 → 4(0) − 2y = 8 → −2y = 8 → divide by −2 → y = −4 → (0, −4).
• x-intercept: y = 0 → 4x − 2(0) = 8 → 4x = 8 → divide by 4 → x = 2 → (2, 0).

🎯 The cover-up method for standard form

🎯 What standard form looks like

General (standard) form: equations written as ax + by = c, where a, b, and c are coefficients or values.

• Example: 7x − 3y = 21 or 2x + 3y = −6.
• The excerpt calls this "general form" and notes it always has "coefficient times x plus (or minus) coefficient times y equals value."

🖐️ How the cover-up method works

• Key insight: at the y-intercept, x = 0, so any term containing x effectively disappears; at the x-intercept, y = 0, so any term containing y disappears.
• To find the y-intercept: physically cover up (or ignore) the x-term, then solve the remaining equation for y.
• To find the x-intercept: cover up the y-term, then solve the remaining equation for x.
• The excerpt suggests using a finger to cover up terms.

🧮 Cover-up examples

Example: 7x − 3y = 21

• y-intercept (cover the 7x term): −3y = 21 → y = −7 → (0, −7).
• x-intercept (cover the −3y term): 7x = 21 → x = 3 → (3, 0).

Example: 12x − 10y = −15

• y-intercept: −10y = −15 → y = 1.5 → (0, 1.5).
• x-intercept: 12x = −15 → x = −5/4 = −1.25 → (−1.25, 0).

Example: x + 3y = 6

• y-intercept: 3y = 6 → y = 2 → (0, 2).
• x-intercept: x = 6 → (6, 0).

🌍 Real-world applications

🌍 Using intercepts to model constraints

Scenario: Jessie's barbecue budget

• Jessie has $30 to spend; burgers cost $1.25 each, hot dogs cost $0.75 each.
• Let x = number of burgers, y = number of hot dogs.
• Total cost equation: 1.25x + 0.75y = 30.
• y-intercept (cover 1.25x): 0.75y = 30 → y = 40 → (0, 40) means 0 burgers and 40 hot dogs.
• x-intercept (cover 0.75y): 1.25x = 30 → x = 24 → (24, 0) means 24 burgers and 0 hot dogs.
• Plot these two points and draw a line; the line shows all combinations that cost exactly $30.

🧠 Alternative reasoning without an equation

• The excerpt notes you can find intercepts directly:
  • If Jessie buys only hot dogs: 30 ÷ 0.75 = 40 hot dogs → (0, 40).
  • If Jessie buys only burgers: 30 ÷ 1.25 = 24 burgers → (24, 0).
• This gives the same two intercept points without writing the equation first.

📊 Intercepts and inequalities

• The excerpt mentions that the real problem is an inequality: Jessie can spend up to $30, not necessarily exactly $30.
• The line represents spending exactly $30; the shaded region below the line represents spending less than $30.
• Don't confuse: the intercepts still define the boundary line, even when the problem involves an inequality.

📝 Summary of methods

Method	When to use	Steps
Substitution	Any linear equation	Substitute x = 0 to find y-intercept; substitute y = 0 to find x-intercept
Cover-up	Equations in standard form (ax + by = c)	Cover the x-term and solve for y; cover the y-term and solve for x
Direct reasoning	Real-world budget/resource problems	Calculate maximum of each item if buying only that item

⚠️ When intercepts don't work

• The excerpt asks "Why can't we use the intercept method to graph the following equation? 3(x + 2) = 2(y + 3)"
• This suggests that some equations may not be suitable for the intercept method, though the excerpt does not explain why in detail.
• Recall: horizontal and vertical lines are special cases where one intercept is missing.

🧭 Overview

🧠 One-sentence thesis

A function is a relationship where each input value produces exactly one output value, and function notation provides a clear way to represent, organize, and distinguish multiple equations in real-world situations.

📌 Key points (3–5)

• What a function is: a relationship between two variables where each input has exactly one output.
• Function notation replaces the dependent variable: instead of writing y = 7x − 3, write f(x) = 7x − 3 to label and distinguish equations.
• Domain and range: domain is all possible input values; range is all resulting output values.
• Common confusion: parentheses in f(x) do not mean multiplication—they separate the function name from the independent variable.
• Why it matters: function notation makes it easy to work with multiple equations, generate tables, and represent real-world situations precisely.

🔤 What is a function?

🔤 Definition and structure

Function: a relationship between two variables such that the input value has ONLY one output value.

• A function is a set of ordered pairs where the first coordinate (input) matches with exactly one second coordinate (output).
• The dependent variable (often y) depends on what is substituted for the independent variable (often x).
• Example: Joseph pays $2 per ride at a theme park. The total money spent m depends on the number of rides r, so m = 2r is a function.

🔤 Function notation anatomy

input
  ↓
f(x) = y ← output
︸︷︷︸
function box

• f represents the function name.
• (x) represents the independent variable.
• The parentheses do not mean multiplication; they separate the function name from the variable.
• f(x) is read as "f of x" and replaces y in an equation.

✍️ Writing equations as functions

✍️ Converting standard form to function notation

Standard equation	Function notation	Explanation
y = 7x − 3	f(x) = 7x − 3	Replace y with f(x)
d = 65t	f(t) = 65t	The dependent variable d becomes f(t)
F = 1.8C + 32	f(C) = 1.8C + 32	Replace F with f(C)

• The dependent variable (the one being calculated) is replaced by the function notation.
• The independent variable (the input) goes inside the parentheses.

✍️ Why use function notation?

• Distinguishing multiple equations: If Joseph, Lacy, Kevin, and Alfred all pay $2 per ride, they all have m = 2r. Without labels, we cannot tell whose equation is whose.
• With function notation:
  • J(r) = 2r represents Joseph's total
  • L(r) = 2r represents Lacy's total
  • K(r) = 2r represents Kevin's total
  • A(r) = 2r represents Alfred's total
• Function notation makes it much easier to graph multiple lines and work with several equations simultaneously.

📋 Generating tables from functions

📋 How to create a table of values

• A function is a type of equation, so you can create a table by choosing values for the independent variable and calculating the output.
• Example: Using Joseph's function J(r) = 2r, where r is the number of rides:

r (rides)	J(r) = 2r	Output (cost)
0	2(0)	0
1	2(1)	2
2	2(2)	4
3	2(3)	6
4	2(4)	8
5	2(5)	10
6	2(6)	12

• Negative values do not make sense for number of rides, so they are not included.

📋 Benefits and limitations of tables

Benefits:

• Precise organization of data
• Easy reference for looking up data
• Provides coordinate points that can be plotted to create a graph

Limitations:

• Cannot represent infinite amounts of data
• Does not always show the possibility of fractional values for the independent variable
• The list cannot include every possibility

🎯 Domain and range

🎯 What domain and range mean

Domain: the set of all possible input values for the independent variable.

Range: the values resulting from the substitution of the domain; the set of all possible output values.

• Domain can be expressed in words, as a set, or as an inequality.
• Range depends on what outputs are produced when the domain values are substituted into the function.

🎯 Determining domain and range

Example 1: Joseph's rides

• Function: J(r) = 2r
• Domain: All whole numbers (cannot ride negative or fractional rides)
• Range: All whole numbers (whole numbers times 2 are still whole numbers, just twice as large)

Example 2: Tennis ball bounce

• Function: h(b) = 0.75b, where b is the previous bounce height
• Domain: b ≥ 0 (previous bounce height can be any positive number)
• Range: All positive real numbers (75% of any positive number is still positive; includes decimals and whole numbers)

Example 3: Dustin mowing lawns

• Dustin charges $10 per hour
• Domain: Hours worked (non-negative real numbers)
• Range: Cost (non-negative multiples of 10)

Example 4: Maria tutoring

• Maria charges $25 per hour with a minimum charge of $15
• Domain: Hours worked (non-negative real numbers)
• Range: Cost starting at $15 and increasing by $25 per hour

🎯 Don't confuse domain and range

• Domain = inputs (what you put into the function)
• Range = outputs (what comes out of the function)
• The excerpt explicitly corrects: "True or false. Range is the set of all possible inputs for the independent variable." This is false—range is the set of outputs, not inputs.

📝 Writing function rules from data

📝 From tables to functions

Example 1: CD costs

Number of CDs	2	4	6	8	10
Cost ($)	24	48	72	96	120

• Pattern: You pay $24 for 2 CDs, $48 for 4 CDs, $120 for 10 CDs.
• Each CD costs $12.
• Function rule: Cost = $12 × number of CDs, or f(x) = 12x

Example 2: Absolute value pattern

x	−3	−2	−1	0	1	2	3
y	3	2	1	0	1	2	3

• Pattern: The output values are always the positive outcomes of the input values.
• This relationship is called the absolute value.
• Function rule: f(x) = |x|

📝 Real-world situations as functions

Example: Maya's internet service

• Monthly access fee: $11.95
• Connection fee: $0.50 per hour
• Let x = number of hours Maya spends online in one month
• Let y = Maya's monthly cost
• Total cost = flat fee + (hourly fee × number of hours)
• Function: y = f(x) = 11.95 + 0.50x

Example: Sheri saving for a car

• Current savings: $515.85
• Saves $62 each week
• Function rule: f(x) = 515.85 + 62x, where x is the number of weeks
• Domain cannot be "all real numbers" because:
  • Negative weeks do not make sense
  • Fractional weeks may not make sense depending on the context
  • Domain should be non-negative whole numbers or non-negative real numbers

Example: Solomon's plumbing

• Flat rate: $40
• Hourly rate: $25 per hour
• Function: f(x) = 40 + 25x, where x is hours worked
• For a 3-hour job: f(3) = 40 + 25(3) = 40 + 75 = 115 dollars

🧮 Working with functions

🧮 Finding range from given domain

Example 1: y = x² − 5 when domain is {−2, −1, 0, 1, 2}

• Substitute each domain value:
  • x = −2: y = (−2)² − 5 = 4 − 5 = −1
  • x = −1: y = (−1)² − 5 = 1 − 5 = −4
  • x = 0: y = 0² − 5 = −5
  • x = 1: y = 1² − 5 = −4
  • x = 2: y = 4 − 5 = −1
• Range: {−5, −4, −1}

Example 2: y = 2x − 3/4 when domain is {−2.5, 1.5, 5}

• Substitute each domain value to find the range.

🧮 Generating tables for specific functions

Example: Angie's earnings

• Angie makes $6.50 per hour as a cashier
• Function: f(x) = 6.50x
• Create a table for input values {5, 10, 15, 20, 25, 30} hours

Example: Triangle area

• Area formula: A = (1/2) × base × height
• If height is 8 cm: A = (1/2) × b × 8 = 4b
• Create a table for base values {1, 2, 3, 4, 5, 6} centimeters

Example: Square root function

• Function: f(x) = √(2x + 3)
• Create a table for input values {−1, 0, 1, 2, 3, 4, 5}

🧭 Overview

🧠 One-sentence thesis

Graphing functions on a coordinate plane visualizes the relationship between independent and dependent variables, and the vertical line test determines whether a graphed relation qualifies as a function.

📌 Key points (3–5)

• Graphing functions: plot ordered pairs as (independent value, dependent value) on a Cartesian plane to visualize relationships.
• Discrete vs continuous: connect points with a smooth curve only when fractional values make sense; use scatter plots when the domain is restricted to whole numbers.
• Function definition: a relation where each input (x-value) has exactly one output (y-value).
• Common confusion: a function can have multiple x-values mapping to the same y-value, but NOT one x-value mapping to multiple y-values.
• Vertical Line Test: a quick visual method to determine if a graph represents a function—if any vertical line intersects the graph more than once, it is not a function.

📍 Plotting on the Cartesian plane

📍 How to plot coordinate points

• Start at the origin (where axes intersect).
• The first coordinate (x) represents horizontal distance:
  • Positive x → move right
  • Negative x → move left
• The second coordinate (y) represents vertical distance:
  • Positive y → move up
  • Negative y → move down

🗺️ Quadrants

The coordinate plane divides into four quadrants:

• First quadrant: upper right
• Second quadrant: upper left
• Third quadrant: lower left
• Fourth quadrant: lower right

🔗 Connecting points vs scatter plots

🔗 When to connect points

• Connect points with a smooth curve when the domain includes all real numbers (or fractional values).
• Example: If measuring the area of a square where side length can be any positive real number (x ≥ 0), connect the points and extend with an arrow to show the function continues.

🔗 When to use scatter plots

• Leave points unconnected when the domain is restricted to whole numbers or discrete values.
• Example: Joseph's ride cost function where r represents number of rides—you cannot ride 2.5 rides, so the points remain as a scatter plot.
• Don't confuse: connecting points implies all values between them are valid solutions.

📖 Reading function rules from graphs

📖 How to find the pattern

1. Read several coordinate points from the graph.
2. Create a table of values.
3. Look for a pattern in how the dependent variable changes with the independent variable.

📖 Example approach

• If distance increases by 1.5 feet every second, the pattern shows: Distance = 1.5 × time.
• The function rule becomes f(x) = 1.5x.

🧮 Understanding relations and functions

🧮 Definitions

Relation: a set of ordered pairs.

Function: a relation between two variables such that each input value has EXACTLY one output value.

🧮 Key distinction

• Each x-value must have only one y-value.
• Multiple x-values CAN share the same y-value.
• Example: In a class height relation, multiple students (domain) can have the same height (range), but one student cannot have multiple heights.

🧮 Testing ordered pairs

To determine if a relation is a function:

• Check if any x-coordinate appears more than once with different y-coordinates.
• Example: {(1, 3), (3, 5), (2, 5), (3, 4)} is NOT a function because x = 3 maps to both y = 5 and y = 4.
• Example: {(-3, 20), (-5, 25), (-1, 5), (7, 12), (9, 2)} IS a function because each x-value appears only once.

✅ The Vertical Line Test

✅ The theorem

Part A: A relation is a function if no vertical line intersects the graph in more than one point.

Part B: If a graphed relation does not intersect any vertical line in more than one point, then that relation is a function.

✅ How to apply it

• Draw (or imagine) vertical lines through different parts of the graph.
• If any vertical line touches the graph at two or more points, the relation is NOT a function.
• This is a quick visual alternative to checking all ordered pairs.

✅ Why it works

• A vertical line represents one x-value.
• If it intersects the graph multiple times, that single x-value corresponds to multiple y-values, violating the function definition.
• Example: A circle fails the vertical line test because vertical lines through the middle intersect at two points.

🧭 Overview

🧠 One-sentence thesis

Function notation provides a systematic way to identify input values and evaluate functions by substituting specific values into the independent variable.

📌 Key points (3–5)

• What function notation shows: the input value for the independent variable appears inside the parentheses, making it easy to see what to substitute.
• How to evaluate: replace the variable with the given value and perform the operations in the function rule.
• Rewriting equations: to use function notation, isolate y on one side, then replace "y =" with "f(x) =".
• Common confusion: function notation f(x) doesn't mean "f times x"—it means "the function f evaluated at input x."
• The machine analogy: a function takes an input, performs operations on it, and produces an output.

📥 Reading function notation

📥 What the parentheses tell you

Function notation allows you to easily see the input value for the independent variable inside the parentheses.

• The value inside the parentheses is what you substitute for the variable.
• Example: In f(4), the number 4 is the input value for x.
• The notation makes it clear which value to use without ambiguity.

🔄 How to evaluate a function

• Use the Substitution Property: replace every instance of the variable with the given value.
• Perform the arithmetic operations according to the function rule.
• Example: For f(x) = negative one-half times x squared, evaluating f(4) means substituting 4 for x, giving negative one-half times 16, which equals negative 8.

✏️ Converting to function notation

✏️ Isolating the dependent variable

• To use function notation, the equation must be written in terms of x.
• This means the y-variable must be isolated on one side of the equal sign.
• Once you have "y =", replace it with "f(x) =".

🔢 Step-by-step conversion

Example from the excerpt: Converting 9x + 3y = 6

1. Subtract 9x from both sides to get 3y = 6 - 9x
2. Divide by 3 to isolate y: y = (6 - 9x)/3 = 2 - 3x
3. Replace "y =" with "f(x) =": f(x) = 2 - 3x

🤖 The function machine metaphor

🤖 How the machine works

• Input: You start with some value (the independent variable).
• Process: The machine performs the operations defined by the function rule.
• Output: The result is your answer.

🔧 Example of the machine in action

• For f(x) = 3x + 2, the machine takes some number x, multiplies it by 3, and adds 2.
• When you evaluate f(2), the input is 2, the machine calculates 3(2) + 2, and the output is 8.
• Don't confuse: The machine doesn't change the function rule—it applies the same rule to different inputs.

🧮 Working with different inputs

🧮 Numerical inputs

• Substitute the specific number for every instance of the variable.
• Example: For f(x) = 6x - 36, evaluating f(2) means calculating 6 times 2 minus 36, which equals 12 - 36 = -24.

🔤 Variable inputs

• You can substitute another variable or algebraic expression.
• Example: For f(x) = 6x - 36, evaluating f(p) means substituting p for x, giving f(p) = 6p - 36.
• The result is an expression in terms of the new variable rather than a single number.

🧭 Overview

🧠 One-sentence thesis

Slope quantifies how steeply a line rises or falls, and when applied to real-world quantities it becomes a rate of change that describes how one quantity varies per unit change in another.

📌 Key points (3–5)

• What slope measures: the ratio of vertical distance (rise) to horizontal distance (run) along a line.
• How to calculate slope: use the formula (change in y) divided by (change in x), or (y₂ - y₁) / (x₂ - x₁) for two points.
• Sign matters: positive slope means the line rises moving left to right; negative slope means it falls; zero slope is horizontal; undefined slope is vertical.
• Common confusion: the slope of a line is the same regardless of which direction you travel along it or which two points you choose.
• Real-world application: slope becomes a rate of change (e.g., speed, growth rate) when describing measurable quantities over time or other variables.

📐 Understanding slope as a ratio

📏 The basic definition

Slope = distance moved vertically / distance moved horizontally

• Often written as: Slope = rise / run
• The "rise" is the vertical change; the "run" is the horizontal change.
• Example: A hill with height 3 and horizontal length 4 has slope 3/4 or 0.75.

➕ Positive vs negative slope

• Positive slope: the line goes up as you move from left to right (climbing).
• Negative slope: the line goes down as you move from left to right (descending).
• Don't confuse: if you reverse direction on the same line, the slope stays the same because both rise and run become negative, and the negatives cancel out.

🔢 Calculating slope from points

🧮 The slope formula

For two points (x₁, y₁) and (x₂, y₂):

Slope = (y₂ - y₁) / (x₂ - x₁)

Alternative notation: m = Δy / Δx, where:

• m denotes slope (mathematical convention)
• Δ (delta) means "change"

🔄 Order doesn't matter

• You get the same slope regardless of which point you label as point 1 or point 2.
• Example: For points (1, 2) and (4, 7), the slope is (7 - 2) / (4 - 1) = 5/3.
• Reversing: (2 - 7) / (1 - 4) = -5 / -3 = 5/3 (same result).

📍 Using lattice points

• Easiest to work with points whose coordinates are all integers (lattice points).
• Draw a right triangle with the line as the hypotenuse to visualize rise and run.

↔️ Special cases: horizontal and vertical lines

➖ Horizontal lines (y = constant)

• The y-coordinate never changes as x increases.
• Rise = 0, so slope = 0 / run = 0.
• All horizontal lines have slope of zero.

⬆️ Vertical lines (x = constant)

• The x-coordinate never changes as y increases.
• Run = 0, so slope = rise / 0.
• Division by zero is undefined, so vertical lines have undefined (or infinite) slope.

📊 Slope as rate of change

⚡ What is rate of change

Rate of change: the slope of a function describing real quantities; represents change in one quantity (y) per unit change in another quantity (x).

• Uses the formula m = Δy / Δx where Δ represents actual changes in measurable units.
• Example: A candle burns from 10 inches to 7 inches in 30 minutes → rate = (7 - 10) / (30 - 0) = -3/30 = -0.1 inches per minute.

🔻 Interpreting negative rates

• Negative rate of change means the quantity is decreasing over time.
• Example: The candle's negative rate (-0.1 inches/minute) shows it's getting shorter as it burns.

🐟 Population growth example

• Fish population increases from 370 to 420 over two months.
• Rate = Δy / Δx = (420 - 370) / 2 = 50/2 = 25 fish per month.

🚚 Distance-time graphs and velocity

🛣️ Slope as velocity

• On a distance-time graph, slope represents velocity (speed with direction).
• Velocity = distance / time, so the rate of change is always a velocity.
• Positive velocity: moving away from starting position; negative velocity: returning toward starting position.

📦 Delivery truck example

The excerpt describes a truck's journey with five stages:

• Stage A: Slope = 80 miles / 2 hours = 40 mph (traveling away).
• Stage B: Slope = 0 (stationary for delivery, 1 hour).
• Stage C: Slope = 40 miles / 1 hour = 40 mph (continuing).
• Stage D: Slope = 0 (stationary for delivery, 2 hours).
• Stage E: Slope = (0 - 120) / 2 = -60 mph (returning home faster).

🔄 Speed vs velocity distinction

• Speed: magnitude only (always positive).
• Velocity: includes direction (can be negative).
• Example: The truck's speed is 60 mph on the return trip, but its velocity is -60 mph because it's traveling in the opposite direction.

🧭 Overview

🧠 One-sentence thesis

The slope-intercept form (y = mx + b) allows you to quickly identify a line's slope and y-intercept, which together provide enough information to graph the line and determine whether lines are parallel.

📌 Key points (3–5)

• Slope-intercept form structure: y = mx + b, where m is the slope and (0, b) is the y-intercept.
• How to graph from the form: plot the y-intercept first, then use the slope (rise over run) to find a second point.
• Effect of changing parameters: changing the slope changes the steepness and direction; changing the intercept shifts the line up or down without changing its angle.
• Common confusion: the b-term includes its sign—in y = 0.5x - 3, the intercept is (0, -3), not (0, 3).
• Identifying parallel lines: any two lines with identical slopes are parallel, regardless of their intercepts.

📐 Understanding slope-intercept form

📐 What the form tells you

Slope-intercept form: y = mx + b, where m is the slope and the point (0, b) is the y-intercept.

• This form directly reveals two key pieces of information about a line.
• m represents the slope (rate of change).
• b represents where the line crosses the y-axis.
• Example: in y = 3x + 2, the slope is 3 and the y-intercept is (0, 2).

⚠️ Watch the signs

• The b-term includes the sign of the operator in front of it.
• y = 0.5x - 3 is identical to y = 0.5x + (-3), so b = -3, not +3.
• Don't confuse: the number after the operator is not always positive—you must include the plus or minus sign as part of b.

🔍 Special cases

Equation form	What it means	Slope	Intercept
y = -7x	Can rewrite as y = -7x + 0	-7	(0, 0) passes through origin
y = -4	Can rewrite as y = 0x - 4	0	(0, -4) horizontal line

🎨 Graphing from slope-intercept form

🎨 The step-by-step method

Instead of making a table of values, you can graph directly:

1. Identify the y-intercept b
2. Plot the point (0, b)
3. Identify the slope m
4. Express the slope as a fraction to find rise and run
5. From the intercept, move over (run) and up/down (rise) to find a second point
6. Draw the line through both points

📊 Converting slope to rise and run

• Express the slope as a fraction in simplest form.
• The numerator is the rise (vertical change).
• The denominator is the run (horizontal change).
• Example: slope m = 3 means rise/run = 3/1, so go over 1 unit and up 3 units.
• Example: slope m = -3 means rise/run = -3/1, so go over 1 unit and down 3 units.
• Example: slope m = 0.75 converts to 3/4, so go over 4 units and up 3 units.

🧮 Working with decimal slopes

For slopes like m = -0.375:

• Convert to a fraction: -0.375 = -3/8
• Rise = -3 (down 3 units)
• Run = 8 (over 8 units)
• This makes it easier to plot accurately on a graph.

🔄 How changing parameters affects the graph

📈 Changing the slope

• Positive slopes: lines increase (go up) as you move from left to right.
• Negative slopes: lines decrease (go down) as you move from left to right.
• Greater absolute value of slope: steeper graph.
• Slope of zero: horizontal line.
• The slope controls the angle and direction of the line.

⬆️ Changing the y-intercept

• Changing the intercept shifts the graph up or down.
• The line remains at the same angle (same slope).
• Example: if (1, 2) is on y = 2x, then (1, 5) is on y = 2x + 3 (add 3 to every y-value).
• Example: if (1, 2) is on y = 2x, then (1, -1) is on y = 2x - 3 (subtract 3 from every y-value).
• The shift is vertical only—no horizontal movement.

⫽ Parallel lines

⫽ The parallel line rule

Any two lines with identical slopes are parallel.

• Parallel lines never intersect.
• If two lines have the same slope but different intercepts, they are parallel.
• Example: y = 2x and y = 2x + 3 are parallel because both have slope m = 2.

🧪 Why this works

• If you try to solve y = 2x and y = 2x + 3 simultaneously, you get 2x = 2x + 3.
• Subtracting 2x from both sides gives 0 = 3, which is impossible.
• This means there is no point (x, y) that satisfies both equations.
• No shared point means the lines never intersect, confirming they are parallel.
• Don't confuse: lines can have the same intercept but different slopes—those lines are not parallel; they intersect at the y-axis.

🧭 Overview

🧠 One-sentence thesis

Graphing linear relationships allows you to solve real-world problems by plotting known information and reading unknown values directly from the graph.

📌 Key points (3–5)

• The four-step problem-solving plan: Understand the problem, devise a plan (make a graph), carry out the plan (read the graph), and check/interpret the answer.
• What you need to graph: identify the y-intercept (starting value when x = 0) and the slope (rate of change per unit).
• How to read the graph: draw horizontal or vertical lines from known values to find unknown values where they intersect the graphed line.
• Common confusion: the y-intercept is not always given directly—sometimes you must calculate it from two points or recognize it as the "starting amount."
• Why it matters: graphs provide visual solutions and approximate answers that can be checked with arithmetic.

📋 The Problem-Solving Plan with Graphs

📋 Four steps for graph-based problems

The excerpt presents a structured approach:

1. Understand the Problem: identify what you know and what you want to find; define variables (let x = ..., let y = ...).
2. Devise a Plan—Translate: make a graph showing the relationship; determine the y-intercept and slope.
3. Carry Out the Plan—Solve: use the graph to answer the question by reading values where lines intersect.
4. Look—Check and Interpret: verify the answer using arithmetic or logical reasoning about the original situation.

🔍 Why this structure works

• Each step builds on the previous one: understanding leads to correct variable assignment, which leads to accurate graphing, which leads to reliable answers.
• The "check" step catches errors from misreading the graph or incorrect setup.

🎯 Setting Up the Graph

🎯 Identifying the y-intercept

The y-intercept is the value of y when x = 0; it represents the starting amount before any change occurs.

• Example (cell phone): You pay $60 upfront when you get the phone, so the y-intercept is (0, 60).
• Example (spring with no weight): When weight = 0 lbs, the spring has its natural length; this is the y-intercept you're solving for.
• Example (Aatif's earnings): He starts with a $50 birthday gift before working any hours, so the y-intercept is (0, 50).

Don't confuse: the y-intercept with the first data point given—sometimes the problem gives you a point like (1, 22) and you must recognize that (0, 0) is also a point (as in the reading example).

📈 Identifying the slope

The slope is the rate of change: how much y increases (or decreases) for each 1-unit increase in x.

• Example (cell phone): You pay $40 per month, so the slope is 40 (cost rises by $40 for each month).
• Example (spring): The spring length increases by 6 inches when weight increases by 3 lbs, so the slope is 6 ÷ 3 = 2 inches per pound.
• Example (Christine reading): She reads 22 pages per hour, so the slope is 22.

How to find slope from two points:

• Calculate rise over run: (change in y) ÷ (change in x).
• Example: points (2, 12) and (5, 18) give slope = (18 - 12) ÷ (5 - 2) = 6 ÷ 3 = 2.

🗺️ Plotting the line

• Plot the y-intercept first.
• Use the slope to find a second point: move right by the "run" amount and up (or down) by the "rise" amount.
• Connect the two points with a straight line.

📖 Reading the Graph to Find Answers

📖 Drawing guide lines

The excerpt describes a consistent technique:

• To find y for a given x: draw a vertical line from the x-value on the horizontal axis up to the graphed line, then draw a horizontal line from that intersection to the y-axis and read the value.
• To find x for a given y: draw a horizontal line from the y-value on the vertical axis to the graphed line, then draw a vertical line down to the x-axis and read the value.

Example (cell phone): "We draw a vertical line from 9 months until it meets the graph, and then draw a horizontal line until it meets the vertical axis" to find the cost after 9 months is approximately $420.

Example (Aatif's surfboard): "We draw a horizontal line from $249 on the vertical axis until it meets the graph and then we draw a vertical line downwards until it meets the horizontal axis" to find he needs approximately 31 hours.

🔢 Approximate vs exact answers

• Graphs give approximate answers because you are reading values visually.
• The excerpt consistently says "approximately" when reading from graphs.
• The check step often gives a more precise answer using arithmetic.

Example (Christine reading): The graph shows approximately 4.5 hours; the arithmetic check gives 100 ÷ 22 = 4.54 hours, which is "very close."

✅ Checking Your Answer

✅ Why checking matters

• Reading graphs can introduce small errors.
• Checking confirms that your setup (y-intercept and slope) was correct.
• The excerpt states "The answer checks out" after every example.

✅ How to check

Use the original problem information to calculate the answer directly:

Example	Graph answer	Check method	Check result
Cell phone (9 months)	~$420	$60 + ($40 × 9)	$60 + $360 = $420 ✓
Spring (no weight)	~8 inches	12 inches - (2 lbs × 2 inches/lb)	12 - 4 = 8 inches ✓
Christine reading	~4.5 hours	100 pages ÷ 22 pages/hour	4.54 hours ✓
Aatif's surfboard	~31 hours	($249 - $50) ÷ $6.50/hour	$199 ÷ $6.50 = 30.6 hours ✓

🧮 Alternative check: reasoning through the rate

Example (spring): "The length of the spring goes up by 6 inches when the weight is increased by 3 lbs. To find the length of the spring when there is no weight attached, we can look at the spring when there are 2 lbs attached. For each pound we take off, the spring will shorten by 2 inches."

• This shows you can work backward using the slope to verify the y-intercept.

🔑 Key Patterns in the Examples

🔑 Two common scenarios

1. You have the y-intercept and slope directly: just plot and read (cell phone, Aatif's earnings).
2. You have two points: calculate the slope, plot both points, draw the line, then read the y-intercept or other value (spring, Christine reading).

🔑 When the y-intercept is the answer

• Spring problem: "What is the length of the spring when no weights are attached?" → This is asking for the y-intercept.
• You graph the two given points, extend the line to the y-axis, and read where it crosses.

🔑 When you need to find x or y from the graph

• Cell phone: given x = 9 months, find y (cost).
• Aatif: given y = $249 (target earnings), find x (hours needed).
• The method is the same: draw guide lines to the axis you need.

🧭 Overview

🧠 One-sentence thesis

Given a graph, two points, or a slope and a point, you can systematically write the equation of a line in slope-intercept form by identifying the slope and y-intercept.

📌 Key points (3–5)

• What you need: To write a line equation in slope-intercept form, you need the slope (m) and the y-intercept (b).
• Five-step method: When you have slope and a point (not the y-intercept), substitute into y = mx + b, plug in the point, solve for b, then rewrite the full equation.
• Two-point method: When given two points, first calculate the slope using the slope formula, then apply the five-step method.
• Function notation: y and f(x) are interchangeable; f(x) = mx + b is the same form, and f(a) = c tells you the point (a, c) is on the line.
• Common confusion: Don't confuse the y-intercept (a number or point) with any other point on the line—only the y-intercept can be directly substituted as b.

📐 The slope-intercept formula

📐 What the formula is

Slope-intercept form: y = (slope)x + (y-intercept), or y = mx + b

• m represents the slope of the line.
• b represents the y-intercept (the y-value where the line crosses the y-axis).
• This form explicitly shows both the slope and the y-intercept.

🔢 Direct substitution when you have both

When you already know the slope and the y-intercept, simply substitute them into the formula.

• Example: slope = 4 and y-intercept = -3 → y = 4x - 3
• Example: slope = 3 and y-intercept = 2 (from a graph) → y = 3x + 2

🛠️ Writing equations from slope and one point

🛠️ The five-step process

When you have the slope but the y-intercept is unknown, use a point on the line to find b:

1. Start with the formula: y = mx + b
2. Substitute the slope: Replace m with the given slope value.
3. Plug in the point: Use the ordered pair (x, y) and substitute those values into the equation.
4. Solve for b: Rearrange and solve to find the y-intercept.
5. Rewrite the equation: Substitute both m and b into y = mx + b.

🧮 Worked scenario

• Given: slope = 4 and point (-1, 5)
• Step 1: y = mx + b
• Step 2: y = 4x + b
• Step 3: 5 = 4(-1) + b
• Step 4: 5 = -4 + b → b = 9
• Step 5: y = 4x + 9

⚠️ Don't confuse

The point you're given is not the y-intercept unless its x-coordinate is zero; you must solve for b algebraically.

📍 Writing equations from two points

📍 Find the slope first

When given two points, the first step is to calculate the slope using the slope formula:

slope = (y₂ - y₁) / (x₂ - x₁)

• Choose one point as (x₁, y₁) and the other as (x₂, y₂).
• Example: points (3, 2) and (-2, 4) → slope = (4 - 2) / (-2 - 3) = 2 / -5 = -2/5

📍 Then apply the five-step method

Once you have the slope, pick either of the two points and use the five-step process to find b.

• Example continued: slope = -2/5, using point (-2, 4):
  • y = (-2/5)x + b
  • 4 = (-2/5)(-2) + b
  • 4 = 4/5 + b → b = 16/5
  • Final equation: y = (-2/5)x + 16/5

🔄 Function notation and slope-intercept form

🔄 What f(x) means

• f(x) is another way to write y; the equation f(x) = mx + b is identical to y = mx + b.
• f(x) emphasizes that x is the independent variable: you substitute x-values into the function to calculate y-values.

🔄 Reading points from function notation

• If f(a) = c, this tells you the point (a, c) is on the graph.
• Example: f(-1) = -7 means the point (-1, -7) is a solution to the equation.

🔄 Writing equations with function notation

You can use function notation to provide a point and then apply the five-step method.

• Example: m = 3.5 and f(-2) = 1 means slope = 3.5 and point (-2, 1).
• Apply the five steps: y = 3.5x + b → 1 = 3.5(-2) + b → b = 8 → f(x) = 3.5x + 8

🌍 Real-world applications

🌍 Identifying slope and y-intercept from context

Real-world problems often describe a starting value (y-intercept) and a rate of change (slope).

Scenario	y-intercept (b)	Slope (m)	Equation
Nadia has $200, earns $7.50/hour	200 (initial savings)	7.50 (dollars per hour)	y = 7.5x + 200
Bamboo grows 12 inches/day, reaches 720 inches in 60 days	0 (starts at 0 inches)	12 (inches per day)	y = 12x

🌍 Answering questions with the equation

Once you have the equation, substitute the known value to find the unknown.

• Example: How many hours for Nadia to have $500?
  • 500 = 7.5x + 200 → 7.5x = 300 → x = 40 hours
• Example: How tall is bamboo after 12 days?
  • y = 12(12) = 144 inches

🌍 When you have two data points

Sometimes a real-world problem gives you two observations instead of a rate and starting value; use the two-point method.

• Example: Bungee cord stretches to 265 feet with 100 lbs, 275 feet with 120 lbs.
• Calculate slope: (275 - 265) / (120 - 100) = 10 / 20 = 0.5 feet per pound.
• Use one point to find b, then write the equation.

🧭 Overview

🧠 One-sentence thesis

Standard form provides a systematic way to write linear equations with integer coefficients on both variables on one side and a constant on the other, which makes it particularly useful for solving real-world problems involving multiple quantities with constraints.

📌 Key points (3–5)

• Standard form structure: A linear equation written as Ax + By = C, where A, B, and C are integers and A and B are not both zero.
• Converting to standard form: Use algebraic operations (Distributive Property, moving terms) to rearrange equations so both variables appear on the same side.
• Finding slope and intercept: From Ax + By = C, the slope is -A/B and the y-intercept is C/B.
• Common confusion: Standard form looks different from slope-intercept form (y = mx + b), but both describe the same line—standard form just doesn't show the slope explicitly.
• Real-world applications: Standard form naturally represents situations with multiple items at different prices or rates that sum to a total.

📐 Understanding Standard Form

📝 Definition and structure

Standard form of a linear equation: Ax + By = C, where A, B, and C are integers and A and B are not both zero.

• All three coefficients (A, B, C) must be integers (whole numbers, positive or negative).
• Both variable terms (x and y) appear on the same side of the equation.
• The constant term stands alone on the other side.
• Example: 2x + 3y = 12 or 7x - 3y = 21.

🔄 Why both A and B cannot be zero

• If both A and B were zero, the equation would become 0 = C, which is either always false (if C ≠ 0) or meaningless (if C = 0).
• At least one variable must be present for the equation to represent a line.

🔧 Converting to Standard Form

🔧 From point-slope or slope-intercept form

The process involves algebraic manipulation to move all variable terms to one side:

Example from the excerpt: Rewrite y - 5 = 3(x - 2) in standard form.

• Step 1: Apply Distributive Property → y - 5 = 3x - 6
• Step 2: Move variables to same side → y - y + 1 = 3x - y
• Step 3: Simplify → 1 = 3x - y (or 3x - y = 1)
• Result: A = 3, B = -1, C = 1

🔧 From simple linear equations

Example from the excerpt: Rewrite 5x - 7 = y in standard form.

• Step 1: Add 7 to both sides → 5x = y + 7
• Step 2: Move y to left side → 5x - y = 7
• Result: A = 5, B = -1, C = 7

📊 Extracting Information from Standard Form

📐 Finding the slope

From Ax + By = C, the slope equals -A/B (negative A divided by B).

Example from the excerpt: For 2x - 3y = -8

• A = 2, B = -3
• Slope = -A/B = -2/(-3) = 2/3

📍 Finding the y-intercept

From Ax + By = C, the y-intercept equals C/B.

Example from the excerpt: For 2x - 3y = -8

• C = -8, B = -3
• y-intercept = C/B = -8/(-3) = 8/3

🔍 Why these formulas work

The excerpt derives these by converting standard form to slope-intercept form:

• Start with Ax + By = C
• Subtract Ax from both sides → By = -Ax + C
• Divide by B → y = (-A/B)x + (C/B)
• Compare to y = mx + b to identify slope m = -A/B and y-intercept b = C/B

🌍 Real-World Applications

🛒 Budget and purchasing problems

Standard form naturally models situations where you buy multiple items with a fixed budget.

Example from the excerpt: Nimitha buys oranges at $2/pound and cherries at $3/pound with $12 total.

• Let x = pounds of oranges, y = pounds of cherries
• Equation: 2x + 3y = 12
• If she buys 4 pounds of oranges: 2(4) + 3y = 12 → 3y = 4 → y = 4/3 pounds of cherries

🚶 Distance and rate problems

Standard form works when combining different speeds or rates to cover a total distance.

Example from the excerpt: Jethro skateboards at 7 mph and walks at 3 mph to cover 6 miles total.

• Let x = hours skateboarding, y = hours walking
• Equation: 7x + 3y = 6
• If he skateboards 0.5 hours: 7(0.5) + 3y = 6 → 3y = 2.5 → y = 5/6 hours walking

💡 Why standard form fits these problems

Feature	Why it helps
Both variables on same side	Represents "total contribution" from both quantities
Constant on other side	Represents the constraint (budget, distance, etc.)
Integer coefficients	Match real unit prices or rates

🧭 Overview

🧠 One-sentence thesis

Linear inequalities in two variables are graphed by drawing a boundary line and shading the appropriate half-plane to show all ordered pairs that satisfy the inequality.

📌 Key points (3–5)

• What the graph shows: A boundary line splits the coordinate plane into two half-planes; the solution set is one of those half-planes, shown by shading.
• Solid vs dashed lines: Use a solid line when the inequality includes equals (≤ or ≥); use a dashed line when it does not (< or >).
• Which side to shade: Shade above the line for "greater than" inequalities (> or ≥); shade below for "less than" inequalities (< or ≤).
• Common confusion: Don't mix up when to use solid vs dashed—the equal sign determines whether points on the line itself are solutions.
• Real-world use: Inequalities model constraints like budget limits, weight mixtures, or sales targets where multiple combinations satisfy the condition.

📐 Understanding half-planes and boundary lines

📐 What a half-plane is

Half-plane: Each of the two pieces created when a line splits the coordinate plane.

• When you graph a linear equation, the line divides the plane into two regions.
• A linear inequality requires graphing not just the line but also all the ordered pairs that satisfy the inequality.
• This collection of solutions is called the solution set and is represented by shading one half-plane.

🖊️ Boundary line: solid or dashed

The boundary line is the line you would graph for the equation (y = mx + b).

Inequality symbol	Line type	Meaning
< or >	Dashed	Points on the line are NOT included in the solution
≤ or ≥	Solid	Points on the line ARE included in the solution

• The equal sign determines inclusion: if the inequality allows equality, the boundary is part of the solution.
• Example: y > 2x - 3 uses a dashed line because points where y equals 2x - 3 are not solutions; y ≥ 2x - 3 uses a solid line because those points are solutions.

🎨 Determining which half-plane to shade

🎨 Greater than: shade above

• > (greater than): The solution is the half-plane above the line (boundary not included).
• ≥ (greater than or equal to): The solution is the half-plane above the line plus all points on the line.

🎨 Less than: shade below

• < (less than): The solution is the half-plane below the line (boundary not included).
• ≤ (less than or equal to): The solution is the half-plane below the line plus all points on the line.

🔍 How to remember

• Think of "greater" as "higher up" on the coordinate plane.
• Think of "less" as "lower down" on the coordinate plane.
• Don't confuse: the inequality sign tells you the direction (above or below), while the presence of the equal sign tells you whether to include the boundary.

🛠️ Step-by-step graphing process

🛠️ The three-step method

Step 1: Graph the boundary line using the most appropriate method.

• Slope-intercept form (y = mx + b): use the y-intercept and slope
• Standard form (Ax + By = C): use the x- and y-intercepts
• Point-slope form: use a point and the slope

Step 2: Determine if the line should be solid or dashed.

• If the equal sign is included (≤ or ≥), draw a solid line.
• If the equal sign is not included (< or >), draw a dashed line.

Step 3: Shade the appropriate half-plane.

• Shade above the line if the inequality is "greater than."
• Shade below the line if the inequality is "less than."

📝 Example walkthrough

For the inequality y ≥ 2x - 3:

• The inequality is in slope-intercept form.
• The inequality symbol is ≥, so draw a solid line (the equal sign is included).
• The inequality is "greater than or equal to," so shade the half-plane above the boundary line.
• The shaded region plus the line itself represents all solutions.

🌍 Real-world applications

🌍 Mixture and cost problems

Coffee blend scenario: Mixing two types of coffee beans—one costs $9.00 per pound, another costs $7.00 per pound. Find mixtures where the blend costs $8.50 per pound or less.

• Let x = weight of $9.00 coffee in pounds; let y = weight of $7.00 coffee in pounds.
• The inequality is: 9x + 7y ≤ 8.50
• Graph using intercepts: when x = 0, y = 1.21; when y = 0, x = 0.944.
• Use a solid line and shade below (≤ means "less than or equal to").
• Only graph the first quadrant because weights cannot be negative.
• The shaded region shows all possible mixtures meeting the cost constraint.

🌍 Sales commission problems

Appliance salesman scenario: Julian earns $60 per washing machine and $130 per refrigerator. How many of each must he sell to make $1,000 or more?

• Let x = number of washing machines; let y = number of refrigerators.
• The inequality is: 60x + 130y ≥ 1000
• Graph using intercepts: when x = 0, y = 16.667; when y = 0, x = 7.692.
• Use a solid line and shade above (≥ means "greater than or equal to").
• The shaded region shows all combinations that meet or exceed the $1,000 goal.

↕️ Special cases: horizontal, vertical, and absolute value inequalities

↕️ Vertical and horizontal lines

Vertical line example: x > 4

• On a number line, this is all numbers greater than 4 (open circle at 4).
• On a coordinate plane, x = 4 is a vertical line four units right of the origin.
• Use a dashed line (no equal sign).
• Shade the half-plane to the right (all x-coordinates larger than 4).

Horizontal line example: y ≤ -5

• The line y = -5 is horizontal, five units below the origin.
• Use a solid line (equal sign included).
• Shade the half-plane below the line.

↕️ Absolute value inequalities

For |x| ≥ 2, rewrite as two separate inequalities:

• x ≤ -2 or x ≥ 2
• Graph both: a vertical line at x = -2 and another at x = 2.
• Both lines are solid (equal sign included).
• Shade the plane to the left of x = -2 and the plane to the right of x = 2.
• The solution is the union of both shaded regions.

Don't confuse: absolute value inequalities often produce two separate regions, not a single connected half-plane.

🧭 Overview

🧠 One-sentence thesis

The graphical method solves systems of linear equations by finding the intersection point of two lines plotted on the same coordinate plane, which represents the one solution that satisfies both equations simultaneously.

📌 Key points (3–5)

• What a solution means: A point is a solution to a system only if it lies on both lines at the same time; the intersection point is the unique solution.
• How to solve by graphing: Plot both equations on the same axes using any graphing method (tables, intercepts, slope-intercept), then identify where the lines cross.
• Verification method: Plug the coordinates of the intersection point into both original equations to confirm it satisfies both.
• Common confusion: A point on one line but not the other is NOT a solution—it must work for both equations.
• Limitation of graphing: This method typically gives only approximate solutions and works best when the answer involves integer coordinates.

🔍 Checking if a point is a solution

🔍 What makes a valid solution

A solution to a system of equations is a point that lies on both lines simultaneously—the coordinates must satisfy both equations.

• Geometrically: the point must be on both lines
• Algebraically: when you substitute the x and y values into each equation, both must be true
• A point on only one line is not a solution to the system

✅ Testing ordered pairs

To check if a coordinate point solves the system:

1. Plug the x and y values into the first equation
2. Check if the equation is true
3. Plug the same values into the second equation
4. Check if that equation is also true
5. Only if both check out is the point a solution

Example: For point (2, 7) in the system y = 4x - 1 and y = 2x + 3:

• First equation: 7 = 4(2) - 1 → 7 = 7 ✓
• Second equation: 7 = 2(2) + 3 → 7 = 7 ✓
• Since both are true, (2, 7) is the solution

Don't confuse: If a point fails even one equation, you don't need to check the other—it's already not a solution to the system.

📊 Graphing both lines to find the solution

📊 The intersection point method

The solution to a linear system is the point where the two lines intersect on the coordinate plane.

• Graph both equations on the same axes
• Use any method: table of values, x- and y-intercepts, or slope-intercept form
• The coordinates where the lines cross are the solution

📈 Graphing techniques

Method	When to use	How it works
Table of values	Equations in y = form	Pick x values, calculate y, plot points
Intercepts	Standard form (ax + by = c)	Find where each line crosses x-axis (set y=0) and y-axis (set x=0)
Slope-intercept	y = mx + b form	Start at y-intercept, use slope to find next points

🎯 Reading the solution

• The intersection point gives both x and y values
• Write the solution as an ordered pair (x, y) or as "x = [value], y = [value]"
• Always verify by substituting back into both original equations

🖩 Using technology to solve systems

🖩 Graphing calculator approach

Graphing calculators can find intersection points more accurately than hand-drawn graphs.

Setup steps:

1. Rewrite both equations in y = mx + b form
2. Enter them as Y1 and Y2 in the calculator
3. Press GRAPH to display both lines

🔎 Three methods to find the intersection

Option 1 - TRACE:

• Move the cursor to the intersection point
• Read coordinates from the screen
• Use ZOOM to get closer and improve accuracy

Option 2 - Table:

• View the table of values (2nd GRAPH)
• Scroll until both Y values match at the same X
• That X value is where the lines intersect

Option 3 - Intersect function:

• Use 2nd TRACE and select "intersect"
• Select first curve, then second curve
• Calculator computes the exact intersection point
• This is generally the most accurate method

🌍 Real-world applications

🌍 Racing problem structure

Real-world problems often involve two different situations that need to be compared to find when/where they meet.

Example setup: Peter runs at 5 feet/second with a 20-foot head start; Nadia runs at 6 feet/second starting from zero.

• Define variables: t = time, d = distance
• Write equations using distance = speed × time
• Nadia: d = 6t
• Peter: d = 5t + 20 (the +20 accounts for his head start)

📍 Interpreting the intersection

• The x-coordinate (time) tells when the two situations are equal
• The y-coordinate (distance) tells where they are equal
• In the racing example: they meet at 20 seconds and 120 feet from the start

⚠️ Limitations of the graphing method

Strengths:

• Provides visual representation of the problem
• Shows how the two situations relate
• Good for understanding what's happening

Weaknesses:

• Requires very careful, accurate graphing
• Only practical when solutions are integers or simple fractions
• Usually gives only approximate solutions
• Other methods (substitution, elimination) are needed for exact answers

Don't confuse: Graphing is excellent for visualization and checking reasonableness, but when you need a precise answer—especially with non-integer solutions—algebraic methods are more reliable.

🧭 Overview

🧠 One-sentence thesis

The substitution method solves systems of linear equations by isolating one variable in terms of another and replacing it in the second equation, allowing you to find exact solutions that may be difficult to read from a graph.

📌 Key points (3–5)

• Core technique: Isolate one variable in one equation, then substitute that expression into the other equation to solve for a single variable.
• When it's useful: Substitution works especially well when one variable has a coefficient of 1, and it finds exact solutions (including fractions) that graphing might miss.
• Real-world applications: The method solves practical problems like comparing phone plans, mixing solutions, or analyzing coin collections.
• Common confusion: Don't forget to substitute back—after finding one variable, you must plug it into your expression to find the other variable.
• Setup matters: Converting word problems into equations requires identifying what each variable represents and writing both a quantity equation and a value/concentration equation.

🔧 The substitution method mechanics

🔧 How substitution works

The method relies on the transitive property: if two expressions both equal the same variable, they equal each other.

Steps:

1. Isolate one variable in either equation
2. Substitute that expression into the other equation
3. Solve the resulting single-variable equation
4. Substitute back to find the second variable

Example: In the racing problem, Nadia's equation is d = 6t and Peter's is d = 5t + 20. Since both equal d, we can write 6t = 5t + 20, giving t = 20 seconds, then d = 120 feet.

🎯 Choosing which variable to isolate

Look for the easiest variable to isolate—usually one with a coefficient of 1.

Example: In the system -4x + y = 2 and 2x + 3y = 6, the y in the first equation has coefficient 1, so solve that equation for y first: y = 2 + 4x.

Don't confuse: "Easiest" means least algebraic work, not necessarily the first equation or the x variable.

📐 Working with standard form equations

📐 Isolating variables from standard form

When equations are in standard form (Ax + By = C), you'll need more algebraic steps to isolate a variable.

Example: Starting with 2x + 3y = 3:

• Subtract 3y from both sides: 2x = 3 - 3y
• Divide both sides by 2: x = (1/2)(3 - 3y)
• Now substitute this expression into the second equation

🔢 Handling fractional solutions

The substitution method excels at finding exact fractional answers that would be hard to read from a graph.

Example: The system 2x + 3y = 3 and 2x - 3y = -1 yields x = 1/2 and y = 2/3. Graphing would make it difficult to pinpoint the intersection at (0.5, 0.667) accurately.

🌍 Real-world problem solving

📱 Comparison problems (phone plans, costs)

These problems compare two options with different fixed costs and per-unit rates.

Setup pattern:

• Let x = the independent variable (minutes, months, items)
• Let y = total cost
• Write each plan as y = (rate)x + (fixed cost)

Example: Vendafone charges $20/month + $0.25/minute: y = 0.25x + 20. Sellnet charges $40/month + $0.08/minute: y = 0.08x + 40. Setting them equal: 0.25x + 20 = 0.08x + 40 gives x = 117.65 minutes as the break-even point.

Interpretation tip: After finding where costs are equal, check the graph or logic to determine which option is better above and below that point.

🪙 Mixture problems (coins, solutions, substances)

Mixture problems require two equations: one for total quantity and one for total value/concentration/mass.

Type	First equation	Second equation
Coins	Total number of items	Total value (item value × count)
Chemical solutions	Total volume	Amount of solute (volume × concentration)
Density mixtures	Total volume	Total mass (volume × density)

🧪 Coin problem example

Janine has 7 coins (nickels and dimes) worth 45 cents total.

Setup:

• Let x = number of nickels, y = number of dimes
• Quantity equation: x + y = 7
• Value equation: 5x + 10y = 45 (nickels worth 5¢, dimes worth 10¢)

Solution: Isolate x from equation 1: x = 7 - y. Substitute into equation 2: 5(7 - y) + 10y = 45, which simplifies to y = 4 dimes, then x = 3 nickels.

🧪 Chemical solution example

A chemist needs 500 ml of 15% copper-sulfate solution, mixing 60% and 5% concentrations.

Setup:

• Let x = ml of 60% solution, y = ml of 5% solution
• Volume equation: x + y = 500
• Solute equation: 0.6x + 0.05y = 75 (since 15% of 500 ml = 75 ml solute)

Solution: From equation 1, x = 500 - y. Substitute: 0.6(500 - y) + 0.05y = 75 gives y = 409 ml and x = 91 ml.

Don't confuse: Convert percentages to decimals (60% → 0.6) before writing equations, and remember the second equation multiplies concentration by volume to get amount of solute.

⚠️ Common pitfalls and tips

⚠️ Always substitute back

After solving for one variable, you must substitute that value back into one of your expressions to find the other variable—the problem isn't finished until you have both values.

⚠️ Check your algebra carefully

The excerpt emphasizes being "careful with the algebra"—distribution, sign errors, and arithmetic mistakes are common when working through substitution steps.

⚠️ Define variables clearly

In word problems, explicitly state what x and y represent (e.g., "x = number of nickels" not just "x = nickels") to avoid confusion when setting up equations.

🧭 Overview

🧠 One-sentence thesis

The elimination method solves systems of linear equations by adding or subtracting equations (sometimes after multiplying them by constants) to eliminate one variable, making it possible to solve for the remaining variable.

📌 Key points (3–5)

• Core idea: Combine two equations so that one variable cancels out, leaving a single equation with one unknown.
• When coefficients are opposites: Simply add or subtract the equations directly to eliminate a variable.
• When coefficients don't match: Multiply one or both equations by constants to create matching (or opposite) coefficients, then add or subtract.
• Common confusion: Don't confuse "multiplying an equation" with "solving for a variable"—when you multiply every term in an equation by the same number, the equation remains true and equivalent.
• Method comparison: Elimination works on all systems and is often faster than substitution when coefficients can be easily matched.

🍎 The basic idea: elimination through addition or subtraction

🍎 The apple-and-banana example

The excerpt introduces elimination with a simple scenario:

• One apple plus one banana costs $1.25
• One apple plus two bananas costs $2.00

How elimination works here:

• The second purchase contains exactly one more banana than the first and costs $0.75 more.
• Subtract the first equation from the second to find the difference in cost that corresponds to the difference in items.
• Algebraically: (a + 2b) − (a + b) = 2.00 − 1.25 → b = 0.75
• Once you know one banana costs $0.75, substitute back to find the apple costs $0.50.

Key insight:

The elimination method looks at the sum or difference of two equations to determine a value for one of the unknowns.

➕ Adding equations vertically

The excerpt recommends visualizing addition by stacking equations and adding down the columns:

• Keep x's and y's in their own columns (like adding units and tens in arithmetic).
• Use placeholders like "0y" if needed to keep columns aligned.

Example from the excerpt:

  3x + 2y = 11
+ 5x − 2y = 13
--------------
  8x + 0y = 24

This gives 8x = 24, so x = 3. Then substitute x = 3 back into either original equation to find y.

Why it works: The y-coefficients (+2 and −2) are opposites, so they cancel when added.

🚣 When coefficients are already opposites

🚣 The canoe example (downstream and upstream)

Scenario:

• Andrew paddles downstream at 7 miles per hour (relative to the bank).
• Upstream he travels at 1.5 miles per hour.
• Let x = Andrew's paddling speed in calm water, y = river current speed.

Setting up the system:

• Downstream: x + y = 7 (his speed is boosted by the current)
• Upstream: x − y = 1.5 (the current works against him)

Elimination:

• The y-coefficients are +1 and −1 (opposites).
• Add the two equations: (x + y) + (x − y) = 7 + 1.5 → 2x = 8.5 → x = 4.25
• Substitute x = 4.25 into the first equation: 4.25 + y = 7 → y = 2.75

Answer: Andrew paddles at 4.25 mph; the river flows at 2.75 mph.

Don't confuse: "Downstream speed" is paddling speed plus current; "upstream speed" is paddling speed minus current.

🔢 Multiplying equations to create matching coefficients

🔢 Why multiply equations?

If the coefficients of a variable are not already opposites (or the same), you can multiply one or both equations by a constant to make them match.

Key principle:

When you multiply an equation, you multiply every term in the equation by a fixed amount (a scalar). The equation remains true.

Analogy from the excerpt:

• If 10 apples cost $5, then 30 apples cost $15 (multiply both sides by 3).
• If 3 bananas + 2 carrots cost $4, then 6 bananas + 4 carrots cost $8 (multiply the entire equation by 2).

🔢 Multiplying one equation

Example from the excerpt:

7x + 4y = 17
5x − 2y = 11

• Multiply the second equation by 2 to get matching y-coefficients (±4):
  • 2(5x − 2y) = 2(11) → 10x − 4y = 22
• Now add to the first equation:

    7x + 4y = 17
  +10x − 4y = 22
  -------------
   17x      = 39  (error in excerpt shows 34, but process is clear)
  

• Solve for x, then substitute back to find y.

When to use: When one coefficient is a simple multiple of the other (e.g., 4 and 2, or 6 and 3).

🚣 Anne's rowing problem (multiplying one equation)

Scenario:

• Downstream: 2 minutes to cover 400 yards → 2(x + y) = 400
• Upstream: 8 minutes to cover 400 yards → 8(x − y) = 400
• Distribute: 2x + 2y = 400 and 8x − 8y = 400

Elimination:

• Multiply the first equation by 4 to match the y-coefficients (±8):
  • 4(2x + 2y) = 4(400) → 8x + 8y = 1600
• Add to the second equation:

   8x + 8y = 1600
  +8x − 8y =  400
  ---------------
  16x      = 2000
  

• Solve: x = 125 yards/min (Anne's rowing speed)
• Substitute back: y = 75 yards/min (river speed)

🔢 Multiplying both equations

🔢 When neither coefficient is a simple multiple

Sometimes you need to multiply both equations by different constants to create a common multiple (like finding a common denominator for fractions).

Example from the excerpt (I-Haul truck rental):

• Anne: 3x + 880y = 840 (3 days, 880 miles, $840)
• Andrew: 5x + 2060y = 1845 (5 days, 2060 miles, $1845)
• x = daily rate, y = per-mile rate

Finding a common multiple:

• The x-coefficients are 3 and 5; their lowest common multiple is 15.
• Multiply the first equation by 5: 15x + 4400y = 4200
• Multiply the second equation by −3: −15x − 6180y = −5535
• Add the equations:

   15x + 4400y =  4200
  −15x − 6180y = −5535
  --------------------
       −1780y = −1335
  

• Solve: y = 0.75 (per-mile rate is $0.75)
• Substitute back into the first equation: 3x + 880(0.75) = 840 → 3x = 180 → x = 60

Answer: I-Haul charges $60 per day plus $0.75 per mile.

Why multiply by negative: Multiplying the second equation by −3 (instead of +3) creates opposite coefficients (15 and −15), so they cancel when added.

📊 Comparing solution methods

📊 Method comparison table

Method	Best used when...	Advantages	Comments
Graphing	You don't need an accurate answer	Easier to see number and quality of intersections; fast with a graphing calculator	Can lead to imprecise answers with non-integer solutions
Substitution	You have an explicit equation for one variable (e.g., y = 14x + 2)	Works on all systems; reduces to one variable	May require extra work to isolate a variable first
Elimination by addition/subtraction	Coefficients of one variable already match or are opposites	Easy to combine equations; quick to solve	Not common for coefficients to already match in a given problem
Elimination by multiplication	No variables are explicit and no coefficients match	Works on all systems; makes it possible to eliminate one variable	Often requires more algebraic manipulation upfront

Key takeaway from the excerpt:

This table is only a guide. You might prefer one method over another, or choose based on the specific system. Try to master all techniques and recognize which is most efficient for each case.

📐 Complementary angles example (comparing all methods)

Problem: Two complementary angles (sum = 90°) satisfy: twice angle A is 9° more than three times angle B. Find each angle.

System:

• x + y = 90
• 2x = 3y + 9

Method 1: Graphing

• Convert to y = mx + b form: y = −x + 90 and y = (2/3)x − 3
• Graph shows intersection around x = 55, y = 35, but it's hard to read precisely.
• Conclusion: Graphing by hand is not the best method here.

Method 2: Substitution

• Solve the first equation for y: y = 90 − x
• Substitute into the second: 2x = 3(90 − x) + 9 → 2x = 270 − 3x + 9 → 5x = 279 → x = 55.8°
• Substitute back: y = 90 − 55.8 = 34.2°

Method 3: Elimination (with multiplication)

• Rearrange and multiply the first equation by 2: 2x + 2y = 180
• Rearrange the second: 2x − 3y = 9
• Subtract: (2x + 2y) − (2x − 3y) = 180 − 9 → 5y = 171 → y = 34.2°
• Substitute back: x = 55.8°

Both substitution and elimination worked well. The excerpt notes that even though the system looked ideal for substitution, elimination was also quick once equations were rearranged properly.

🎯 Practice problems overview

🎯 Types of problems in the practice set

The excerpt includes a variety of practice problems (not solved in detail):

• Direct systems: Solve by addition, subtraction, or multiplication.
• Word problems: Candy store purchases, plane speeds with wind, taxi fares, call-box rates, plumber and builder hours, part-time job wages.
• Geometry: Complementary and supplementary angles.
• Mixture/investment: Fertilizer solutions, stock investments.
• Age problems: The excerpt mentions a Khan Academy video on "age problems," a common standardized-test topic.

Recommendation from the excerpt:

Try to master all the techniques, and recognize which one will be most efficient for each system you are asked to solve.

Note on showing work: The excerpt cautions that video narrators are "not always careful about showing work," and advises students to "try to be neater in your mathematical writing."

🧭 Overview

🧠 One-sentence thesis

Linear systems can be classified into three types—consistent (one solution), inconsistent (no solution), or dependent (infinite solutions)—based on whether their lines intersect at one point, are parallel, or are identical.

📌 Key points (3–5)

• Three system types: consistent systems have exactly one solution, inconsistent systems have no solutions, and dependent systems have infinitely many solutions.
• Geometric interpretation: lines that intersect at one point are consistent; parallel lines (never intersecting) are inconsistent; identical lines are dependent.
• Algebraic identification: different slopes → consistent; same slope but different y-intercepts → inconsistent; same slope and same y-intercept → dependent.
• Common confusion: dependent systems have infinite solutions, but not any point works—only points that lie on the shared line satisfy both equations.
• Solution method clues: solving algebraically yields one answer (consistent), a false statement like 0 = 13 (inconsistent), or a true statement like 9 = 9 (dependent).

📐 Identifying system types by slope and intercept

📏 Slope-intercept comparison method

To classify a system without graphing, rewrite both equations in slope-intercept form (y = mx + b) and compare:

Condition	Slopes	Y-intercepts	System type	Number of solutions
Lines intersect	Different	Any	Consistent	Exactly one
Lines parallel	Same	Different	Inconsistent	None
Lines identical	Same	Same	Dependent	Infinitely many

🔍 Why slopes and intercepts matter

• Different slopes: lines must cross somewhere because they tilt at different angles.
• Same slope, different intercepts: lines run parallel and never meet.
• Same slope and intercept: the two equations describe the exact same line.

📝 Example: consistent system

Given:

• 2x − 5y = 2 → y = (2/5)x − 2/5
• 4x + y = 5 → y = −4x + 5

Slopes are 2/5 and −4, which are different → the system is consistent (one solution).

📝 Example: inconsistent system

Given:

• 3x = 5 − 4y → y = (−3/4)x + 5/4
• 6x + 8y = 7 → y = (−3/4)x + 7/8

Slopes are both −3/4, but y-intercepts differ (5/4 vs 7/8) → the system is inconsistent (no solution).

📝 Example: dependent system

Given:

• x + y = 3 → y = −x + 3
• 3x + 3y = 9 → y = −x + 3

Both slope and y-intercept are identical → the system is dependent (infinite solutions).

🧮 Identifying system types algebraically

🔢 What happens when you solve

Instead of comparing slopes, you can solve the system and interpret the result:

• One unique answer (e.g., x = 15/8, y = 21/4) → consistent system.
• A false statement (e.g., 0 = 13) → inconsistent system.
• A true statement (e.g., 9 = 9) → dependent system.

🚫 Inconsistent systems yield false statements

Example:

• 3x − 2y = 4
• 9x − 6y = 1

Multiply the first equation by 3 → 9x − 6y = 12. Subtracting the second equation gives 0 = 13, which is never true → inconsistent.

Why this happens: the equations describe parallel lines that never meet, so no (x, y) pair can satisfy both.

✅ Dependent systems yield true statements

Example:

• 4x + y = 3
• 12x + 3y = 9

Solve the first for y: y = −4x + 3. Substitute into the second: 12x + 3(−4x + 3) = 9 → 9 = 9, always true → dependent.

Why this happens: the second equation is just three times the first, so they trace the same line.

⚠️ Don't confuse infinite solutions with "any solution"

A dependent system has infinitely many solutions, but only points on the shared line satisfy both equations.

• Example: for 4x + y = 3 and 12x + 3y = 9, the point (1, −1) works, and so does (−1, 7).
• But (3, 5) does not work—it doesn't lie on the line.
• For every x-value there is exactly one y-value that fits, just as for a single line.

🌍 Real-world applications

🎬 Consistent system: movie rental membership

Scenario: CineStar offers two plans:

• Membership: $45/year + $2 per movie
• No membership: $0/year + $3.50 per movie

Equations:

• y = 45 + 2x (membership)
• y = 3.5x (no membership)

Solution: Set them equal: 3.5x = 45 + 2x → 1.5x = 45 → x = 30 movies.

Interpretation: after renting 30 movies, membership becomes cheaper. The lines cross because the per-movie cost (slope) differs.

🚫 Inconsistent system: competing rental stores

Scenario: Two stores both charge $3 per movie, but different memberships:

• Movie House: $30/year + $3 per movie
• Flicks for Cheap: $15/year + $3 per movie

Equations:

• y = 30 + 3x
• y = 15 + 3x

Solution: Substitute: 15 + 3x = 30 + 3x → 15 = 30, always false → inconsistent.

Interpretation: Movie House will never be better because it costs more upfront and the same per movie. The lines are parallel (same slope, different intercepts).

🔁 Dependent system: fruit purchase

Scenario:

• Peter buys 2 apples + 3 bananas for $4.
• Nadia buys 4 apples + 6 bananas for $8.

Equations:

• 2a + 3b = 4
• 4a + 6b = 8

Solution: Multiply the first by −2 and add: 0 = 0, always true → dependent.

Interpretation: Nadia's purchase is exactly double Peter's, so the second equation gives no new information. We cannot determine individual fruit prices—infinitely many price combinations fit both statements.

🧭 Overview

🧠 One-sentence thesis

A systematic four-step problem-solving plan—understand, devise a plan, carry out the plan, and check—equips students with multiple strategies to tackle real-world mathematical problems effectively.

📌 Key points (3–5)

• The four-step plan: understand the problem, make a plan (translate), solve (carry out), and check/interpret the results.
• Multiple strategies available: drawing diagrams, making tables, looking for patterns, guess-and-check, working backwards, using formulas, writing equations, and more.
• Strategies often combine: most problems benefit from using several strategies together (e.g., drawing a diagram + looking for patterns, or making a table + drawing a graph).
• Common confusion: students may rush to solve without fully reading or understanding the problem; always read carefully and highlight key words before starting.
• Checking is essential: substitute your solution back into the original problem to verify it makes sense and produces a true statement.

📖 Understanding the problem (Step 1)

📖 What to do first

Before attempting any solution, you must extract the right information from the problem statement. The excerpt provides a checklist:

• Read carefully and completely: many mistakes come from not reading all sentences.
• Highlight key words: mathematical operations (sum, difference, product), mathematical verbs (equal, more than, less than, is), and nouns (time, distance, people).
• Ask if you've seen something similar: the specific nouns and verbs may differ, but the general situation might match a problem you've solved before.
• Identify the question: what are you being asked to find?
• List the given facts: typically numbers or other concrete information.

🔤 Choosing variables

Once you understand the problem, declare what variables will represent the unknown quantities. The excerpt recommends using letters that make sense (e.g., t for time, d for distance).

🛠️ Making a plan (Step 2)

🛠️ Strategy toolbox

The excerpt lists common strategies you can use:

Strategy	Description
Drawing a diagram	Visual representation of the problem
Making a table	Organizing data in rows and columns
Looking for a pattern	Identifying repeating or predictable sequences
Guess and check	Trying a solution and refining based on results
Working backwards	Starting from the result and reversing operations
Using a formula	Applying a known mathematical relationship
Reading/making graphs	Visual data representation
Writing equations	Translating words into algebraic expressions
Using linear models	Applying linear relationships
Dimensional analysis	Converting units systematically
Using the right function type	Matching the situation to the appropriate function

🔗 Combining strategies

In most problems, you will use a combination of strategies.

• Drawing a diagram and looking for patterns work well together for most problems.
• Making a table and drawing a graph are often paired.
• Writing equations is the most frequent strategy in algebra.

Don't confuse: you don't need to pick just one strategy; effective problem-solving often requires layering multiple approaches.

✅ Solving and checking (Step 3 & 4)

✅ Carry out the plan

Once you have a strategy, execute it systematically. The excerpt emphasizes that this step should follow naturally from your plan in Step 2.

✅ Check and interpret

Always verify your answer using these questions:

• Does the answer make sense? Consider whether the magnitude and sign are reasonable for the context.
• Does substitution work? Plug your solution back into the original problem to see if it produces a true statement.
• Can you use another method? Arriving at the same answer via a different strategy increases confidence.

Example: In the corn harvest problem (Example 2), the crew harvests 198,000 ears in 20 hours, yielding 9,900 ears per hour. Check: 9,900 × 20 = 198,000 ✓

🔄 Comparing alternative approaches (Step 4 extended)

🔄 Why compare methods

The excerpt notes that most problems can be solved using several different strategies. When you know all the strategies, you can:

• Choose methods you're most comfortable with.
• Select approaches that make the most sense for the specific problem type.
• Understand the strengths and weaknesses of each strategy.

🔄 Flexibility in problem-solving

The excerpt states that the book will often demonstrate more than one method for the same problem, helping students see which strategies work best in different contexts.

Key insight: there is rarely one "correct" strategy; effective problem-solvers develop judgment about which tools to use when.

🧮 Worked examples

🧮 Age problem (Example 1)

Problem: Jeff is 10 years old, Ben is 4. When will Jeff be twice as old as Ben?

Solution approach: Look for a pattern by doubling Ben's possible ages:

• Ben age 4 → Jeff would be 8 (but Jeff is already 10, so this is in the past)
• Ben age 5 → Jeff would be 10 (doesn't work; Jeff is already 10 now)
• Ben age 6 → Jeff would be 12 (in 2 years, this works!)

Answer: Jeff will be 12 years old.

🧮 Corn harvest problem (Example 2)

Problem: A field has 660 rows with 300 ears per row. The crew will harvest in 20 hours. How many ears per hour?

Solution approach: Use reasoning and arithmetic:

1. Total ears: 660 × 300 = 198,000
2. Ears per hour: 198,000 ÷ 20 = 9,900

Answer: The crew harvests 9,900 ears per hour.

Don't confuse: the problem asks for rate (ears per hour), not total ears; division is needed after finding the total.

📝 Four-step summary

The excerpt concludes with a condensed version of the plan:

1. Understand the problem
2. Devise a plan – Translate: set up an equation, draw a diagram, make a chart, or construct a table
3. Carry out the plan – Solve
4. Check and Interpret: verify you used all information and that the answer makes sense

This framework applies to all mathematical problem-solving, not just algebra, and becomes more powerful as you add more strategies to your toolbox.

🧭 Overview

🧠 One-sentence thesis

Organizing data into tables and identifying patterns are two fundamental problem-solving strategies that make numerical relationships easier to see and use for finding solutions.

📌 Key points (3–5)

• When to use tables: Best when a problem has data that needs organizing or requires recording large amounts of information.
• When to look for patterns: Effective when a situation has a readily apparent pattern that can be extended to find the solution.
• Both strategies work together: Tables often help reveal patterns; patterns can be verified by organizing data in tables.
• Common confusion: Don't confuse "making a table" with just listing numbers—the table should organize information to reveal relationships.
• Multiple approaches possible: The same problem can often be solved using either strategy, and comparing methods helps verify answers.

📊 Making tables to organize information

📊 When tables are most useful

Tables are highly effective when:

• A problem contains data that needs to be organized
• You need to record a large amount of information
• You want to make patterns and numerical relationships easier to see

A table is a highly effective problem-solving strategy when data needs to be organized.

🏃 Example: Jogging schedule problem

The excerpt shows a problem where Josie increases her jogging time by 2 minutes per day each week, jogging 6 days per week.

How the table helps:

• Organizes weeks in columns (Week 1, Week 2, Week 3, Week 4)
• Shows daily minutes (10, 12, 14, 16)
• Shows weekly totals (60, 72, 84, 96 minutes per week)
• Reveals the pattern: weekly total increases by 12 minutes each week

Finding the solution:

• The pattern shows adding 12 minutes per week
• Continue the pattern to week 6: 120 minutes total
• Can verify with equation: t = 60 + 12(w - 1)

✅ Checking your work

After using a table to find a pattern:

• Verify the pattern starts correctly
• Check that the pattern rule applies consistently
• Substitute your answer back into any equation to confirm

🔍 Looking for patterns

🔍 What "readily apparent pattern" means

A readily apparent pattern means the pattern is easy to see.

When a pattern is obvious, you may not need a table—you can use the pattern directly to arrive at your solution.

🎾 Example: Tennis ball triangle

The excerpt describes arranging tennis balls in triangular layers.

How the pattern works:

• One layer: 1 ball
• Two layers: add balls from top layer to bottom layer (1 + 2 = 3)
• Three layers: 1 + 2 + 3 = 6
• Four layers: 6 + 4 = 10

The pattern rule:

• Each layer has one more ball than the previous layer
• Each layer number equals the number of balls in that layer
• To find 8 layers: continue adding (10 + 5 = 15, then 15 + 6 = 21, then 21 + 7 = 28, then 28 + 8 = 36)

Verification:

• Check by adding all layers: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36 balls
• The answer checks out

🔄 Pattern characteristics

When looking for patterns, notice:

• How values change from one step to the next
• Whether the change is consistent (adding the same amount, multiplying by the same factor)
• Whether you can describe the rule in words before applying it

🔄 Comparing the two strategies

💵 Example: Money problem with two approaches

Problem: Andrew cashes a $180 check and wants $10 and $20 bills. The teller gives him 12 bills total. How many of each?

Method 1: Making a table

Tens	0	2	4	6	8	10	12
Twenties	9	8	7	6	5	4	3

• Find the combination that sums to 12 bills
• Answer: six $10 bills and six $20 bills

Method 2: Using a pattern

• Start with the most $20 bills possible
• Pattern: for every pair of $10 bills added, reduce $20 bills by one
• Answer: 6($10) + 6($20) = $180

Don't confuse: Both methods reach the same answer but organize thinking differently—tables show all possibilities; patterns focus on the relationship.

🎓 Example: Homecoming parade

Problem: One kindergartener, two first-graders, three second-graders, through 12th grade march. How many students total?

Solution 1 (table):

• List grades K through 12
• List number of students: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13
• Sum all numbers = 91 students

Solution 2 (pattern):

• Pattern: number of students is one more than grade level
• Sum numbers from 1 (kindergarten) through 13 (12th grade)
• Answer: 91 students

🎯 When to choose which strategy

• Use a table when you need to see multiple possibilities side by side
• Use a pattern when the relationship is clear and extends predictably
• Use both to verify your answer—if both methods give the same result, you can be confident

🛠️ The four-step problem-solving plan

🛠️ The framework for all strategies

Every problem should follow these steps:

1. Understand the problem – Read carefully and identify what you need to find
2. Devise a plan – Translate – Choose a strategy (table, pattern, equation, diagram, chart)
3. Carry out the plan – Solve – Execute your chosen strategy
4. Check and Interpret – Verify you used all information and the answer makes sense

🔗 How tables and patterns fit

• Both are strategies you choose in Step 2 (Devise a plan)
• Both help you carry out Step 3 (Solve)
• Both provide ways to verify your work in Step 4 (Check)

Key insight: These are the most common strategies used before learning algebra, and they continue to be useful alongside algebraic methods.

🧭 Overview

🧠 One-sentence thesis

The "guess and check" and "work backwards" strategies provide systematic ways to solve problems by either refining guesses until you reach the correct answer or reversing operations from a known result to find an unknown starting value.

📌 Key points (3–5)

• Guess and check: make an initial guess, test it, then adjust based on whether the result is too big or too small; patterns often emerge to guide better guesses.
• Work backwards: start with the final result and reverse the operations in opposite order to find the unknown starting value.
• When to use each: guess and check works well when you need two related numbers; work backwards suits problems with a series of operations applied to an unknown.
• Common confusion: both strategies can solve the same problem, but one is often more straightforward—choosing the right approach depends on the problem structure.
• Real-world application: these strategies solve everyday problems like splitting lengths, tracking bank balances, and comparing rental costs.

🎯 The Guess and Check Strategy

🔍 How the method works

The "guess and check" method: guess a solution, test it in the problem, and adjust the guess based on whether the answer is too big or too small until you arrive at the correct solution.

• You don't need the perfect answer immediately; each guess gives you information.
• The process reveals patterns that help you make smarter guesses.
• Keep refining until the guess satisfies all conditions.

📏 Example: Cutting a ribbon

Problem: Nadia cuts a 48-inch ribbon into two pieces; one piece is three times as long as the other. How long is each piece?

Solution process:

• Need two numbers that add to 48, where one is three times the other.
• First guess: 5 and 15 → sum is 20 (too small).
• Second guess: 6 and 18 → sum is 24 (still too small, but exactly half of 48).
• Pattern noticed: multiply both by 2.
• Third guess: 12 and 36 → sum is 48 ✓ Correct.

Key insight: The second guess being exactly half of the target revealed a doubling pattern, making the next guess straightforward.

🚗 Example: Comparing rental costs

Problem: Hana's rental costs $50/day + $0.40/mile; Peter's costs $70/day + $0.30/mile. At how many miles do they pay the same?

Solution process:

• Guess 50 miles → Hana: $70, Peter: $85 (difference: $15).
• Guess 60 miles → Hana: $74, Peter: $88 (difference: $14).
• Pattern: 10 more miles reduced the gap by $1; need to close a $14 gap → try 140 more miles.
• Guess 200 miles → Hana: $130, Peter: $130 ✓ Correct.

Key insight: Tracking how the difference changes between guesses helps you calculate the jump needed to reach equality.

⏮️ The Work Backwards Strategy

🔄 How the method works

The "work backwards" method: start with the final result and apply all operations in reverse order until you find the unknown starting value.

• Works best when a series of operations is applied to an unknown and you know the end result.
• Reverse each operation: addition becomes subtraction, multiplication becomes division, and vice versa.
• Apply reversals in the opposite sequence.

💰 Example: Bank account balance

Problem: Anne writes a check for $24.50, withdraws $80, deposits $235, and ends with $451.25. How much did she start with?

Solution process:

• End balance: $451.25.
• Reverse the deposit: subtract $235.
• Reverse the withdrawals: add back $80 and $24.50.
• Calculation: 451.25 − 235 + 80 + 24.50 = 320.75.
• Anne started with $320.75.

Why it works: Each operation is undone in reverse order, peeling back the changes to reveal the original amount.

👨‍👧 Example: Finding Nadia's age

Problem: Nadia's father is 36, which is 16 years older than four times Nadia's age. How old is Nadia?

Work backwards approach:

• Father's age: 36.
• Reverse "add 16": subtract 16 → 36 − 16 = 20.
• Reverse "multiply by 4": divide by 4 → 20 ÷ 4 = 5.
• Nadia is 5 years old.

Comparison note: The excerpt mentions this problem can also be solved by guess and check, but work backwards is more direct here.

🔀 Comparing and Choosing Strategies

🧩 When each strategy fits best

Strategy	Best for	Example from excerpt
Guess and check	Finding two related unknowns; problems where patterns emerge	Ribbon cutting (two pieces with a 3:1 ratio)
Work backwards	Series of operations on one unknown with known result	Bank balance (multiple transactions, known ending)

🎯 Same problem, different paths

The excerpt shows that Nadia's age problem can be solved both ways:

• Guess and check: Guess Nadia's age, calculate 4 × age + 16, see if it equals 36, adjust.
• Work backwards: Start with 36, subtract 16, divide by 4.

Key takeaway: Work backwards is more straightforward when you have a clear chain of operations; guess and check is useful when the relationship between unknowns is less direct.

🚫 Don't confuse

• Guess and check is not random guessing—you use each result to inform the next guess.
• Work backwards is not just "doing the opposite"—you must reverse both the operations and their order.

🛠️ The Four-Step Problem-Solving Plan

📋 The framework

Every problem in this section follows the same structure:

1. Understand the problem: identify what you know and what you need to find.
2. Devise a plan – Translate: choose a strategy (guess and check, work backwards, etc.).
3. Carry out the plan – Solve: execute the strategy step by step.
4. Look – Check and Interpret: verify the answer makes sense in the original problem.

✅ Why checking matters

• In the ribbon example, checking means verifying that 12 + 36 = 48 and that 36 = 3 × 12.
• In the bank example, checking means applying the original operations forward: 320.75 − 24.50 − 80 + 235 = 451.25.
• The check step catches errors and confirms the solution satisfies all conditions.

🧭 Overview

🧠 One-sentence thesis

Graphing linear relationships allows you to solve real-world problems by plotting known information and reading unknown values directly from the graph.

📌 Key points (3–5)

• Core strategy: translate word problems into graphs with meaningful axes (e.g., time vs. cost, weight vs. length), then read the answer visually.
• Two common graph types: slope-intercept graphs (when you know initial value and rate of change) and two-point graphs (when you know two specific data pairs).
• How to read the answer: draw vertical or horizontal lines from the known value to the graph, then to the other axis to find the unknown.
• Common confusion: the y-intercept is not always zero—it represents the starting value when x = 0 (e.g., upfront cost, original length with no weight).
• Always verify: after reading the graph, check your answer by recalculating with the original problem's numbers and logic.

📐 The Four-Step Problem-Solving Plan

📐 Step 1: Understand the problem

• Identify what you know (given values, rates, initial conditions).
• Identify what you want (the unknown quantity).
• Define variables clearly:
  • Let x = the independent variable (often time, weight, or quantity).
  • Let y = the dependent variable (often cost, length, or pages).

📐 Step 2: Devise a plan—Translate

• Decide which axis represents which variable.
• Determine key graph features:
  • Y-intercept: the value of y when x = 0 (starting point, initial condition).
  • Slope: the rate of change (how much y changes per unit of x).
• Choose your graphing method:
  • Slope-intercept method: plot the y-intercept, then use the slope to find other points.
  • Two-point method: plot two known (x, y) pairs, draw a line through them, and extend it.

📐 Step 3: Carry out the plan—Solve

• Draw the graph with labeled axes.
• To find an unknown:
  • Draw a vertical line from the known x-value to the graph, then a horizontal line to the y-axis (to find y).
  • Draw a horizontal line from the known y-value to the graph, then a vertical line to the x-axis (to find x).
• Read the approximate value from the graph.

📐 Step 4: Look—Check and Interpret

• Recalculate using the problem's logic (e.g., multiply rate by time, add initial cost).
• Compare the calculated answer to the graph reading.
• If they match (or are very close), the answer checks out.

📱 Slope-Intercept Graphing

📱 When to use it

• Use this method when you know:
  • An initial value (the y-intercept, when x = 0).
  • A constant rate (the slope, how much y changes per unit of x).

📱 Example: Cell phone deal

Problem: A phone costs $60 upfront, and the plan costs $40 per month. How much after 9 months?

• Step 1: Let x = months, y = total cost. Know: $60 initial, $40/month.
• Step 2: Y-intercept = (0, 60); slope = 40 (cost rises $40 per month).
• Step 3: Graph the line. Draw a vertical line from x = 9 to the graph, then horizontal to the y-axis → approximately $420.
• Step 4: Check: $60 + ($40 × 9) = $60 + $360 = $420. ✓

Don't confuse: The y-intercept is the upfront cost, not zero—you pay $60 even before the first month.

📱 Example: Surfboard savings

Problem: Aatif has $50, earns $6.50/hour, needs $249 for a surfboard. How many hours to work?

• Step 1: Let x = hours, y = earnings. Know: starts with $50, earns $6.50/hour, needs $249.
• Step 2: Y-intercept = (0, 50); slope = 6.5.
• Step 3: Graph the line. Draw a horizontal line from y = 249 to the graph, then vertical to the x-axis → approximately 31 hours.
• Step 4: Check: needs $249 − $50 = $199. $199 ÷ $6.50/hour ≈ 30.6 hours. ✓

📏 Two-Point Graphing

📏 When to use it

• Use this method when you know two specific (x, y) pairs but not the y-intercept or slope directly.
• Plot both points, draw a line through them, and extend it to find other values.

📏 Example: Spring length

Problem: A spring is 12 inches with 2 lbs attached, 18 inches with 5 lbs. What is the length with 0 lbs (no weight)?

• Step 1: Let x = weight (lbs), y = length (inches). Know: (2, 12) and (5, 18). Want: length when x = 0.
• Step 2: Plot (2, 12) and (5, 18). Draw a line through them and extend it.
• Step 3: Find the y-intercept (where the line crosses the y-axis when x = 0) → approximately 8 inches.
• Step 4: Check: slope = (18 − 12) / (5 − 2) = 6 inches / 3 lbs = 2 inches/lb. From 2 lbs to 0 lbs, remove 2 lbs → spring shortens by 2 × 2 = 4 inches. 12 − 4 = 8 inches. ✓

Key insight: The y-intercept (8 inches) is the original length of the spring with no weight—this is what the problem asked for.

📏 Example: Reading pages

Problem: Christine reads 22 pages in 1 hour. How long to read 100 pages at the same rate?

• Step 1: Let x = hours, y = pages. Know: (1, 22) and (0, 0) (0 hours = 0 pages).
• Step 2: Plot (0, 0) and (1, 22). Draw a line through them.
• Step 3: Draw a horizontal line from y = 100 to the graph, then vertical to the x-axis → approximately 4.5 hours.
• Step 4: Check: rate = 22 pages/hour. 100 pages ÷ 22 pages/hour ≈ 4.54 hours. ✓

Don't confuse: The point (0, 0) is valid here because reading 0 hours means 0 pages read—this is not always the case (e.g., the phone example had a non-zero y-intercept).

🔍 Reading and Interpreting Graphs

🔍 How to read a graph for unknowns

To find	Draw from	Then draw to	Read from
y (given x)	Vertical line from x-value	Horizontal line to y-axis	y-axis
x (given y)	Horizontal line from y-value	Vertical line to x-axis	x-axis
y-intercept	—	—	Where line crosses y-axis (x = 0)

🔍 Why graphs work for linear problems

• The excerpt states that "within certain weight limits" or under constant conditions, relationships are linear functions.
• Linear means the rate of change (slope) is constant.
• Example: Christine reads at a "constant rate of pages per hour"—this makes the relationship linear and graphable.

🔍 Approximation vs. exact values

• Graphs give approximate answers because you are reading visually.
• The excerpt consistently says "approximately" (e.g., "approximately $420," "approximately 4.5 hours").
• Always check with exact arithmetic to confirm the graph reading is close enough.

✅ Verification Strategies

✅ Why Step 4 matters

• Graphs can be misread (e.g., wrong scale, imprecise drawing).
• Verification ensures the answer makes sense and matches the problem's logic.

✅ Common verification methods

• Multiply rate by quantity: e.g., $40/month × 9 months = $360, plus $60 upfront = $420.
• Divide total by rate: e.g., $199 needed ÷ $6.50/hour ≈ 30.6 hours.
• Use slope to backtrack: e.g., slope = 2 inches/lb; removing 2 lbs shortens spring by 4 inches.

✅ What "checks out" means

• The calculated answer and the graph reading should be very close (within rounding error).
• Example: graph shows ~4.5 hours, calculation gives 4.54 hours → close enough, answer checks out.

🧭 Overview

🧠 One-sentence thesis

Using formulas as a problem-solving strategy allows you to translate real-world situations into mathematical equations and solve for unknown dimensions, quantities, or values.

📌 Key points (3–5)

• When to use formulas: Problems involving geometric shapes, rates, conversions, or relationships that fit known mathematical formulas.
• The translation step: Convert the word problem into an algebraic equation by identifying what the formula variables represent in the situation.
• Substitution and solving: Replace known values into the formula and use algebra to isolate the unknown variable.
• Common confusion: Don't confuse the formula variables with the problem variables—you may need to express one variable in terms of another based on the problem's constraints.
• Verification matters: Check your answer by substituting back into the original situation to ensure it makes sense.

🔧 The Formula Strategy in Action

🏗️ Core approach

The excerpt demonstrates solving problems by:

• Identifying an appropriate formula (area, volume, conversion rates, etc.)
• Translating the problem's constraints into algebraic relationships
• Substituting known values and solving for unknowns

📐 The architect example

Problem: A room will be twice as long as it is wide, with total area of 722 square feet. Find the dimensions.

Solution process:

• Start with the rectangle area formula: Area = length × width
• Translate "twice as long as wide" into: length = 2 × width
• Substitute: Area = (2w) × w = 2w²
• Plug in known area: 722 = 2w²
• Solve: w² = 361, so w = 19 feet
• Therefore: width = 19 feet, length = 38 feet

Key insight: The problem gave a relationship between two variables, allowing you to express everything in terms of one unknown.

📏 Types of Formulas Used

📦 Geometric formulas

The excerpt references several spatial formulas:

Formula type	Expression	Variables
Rectangle area	A = length × width	A = area, l = length, w = width
Cone volume	Volume = π × r² × h ÷ 3	r = radius, h = height
Cube surface area	Surface Area = 6x²	x = side length
Circle area	(implied for DVD storage)	Between two radii

🔄 Rate and conversion formulas

• Unit conversions: 1 meter = 39.37 inches; 1 mph = 0.44704 meters/second
• Linear rates: Thickness per page, cost per hour, speed calculations
• Proportional relationships: If 550 pages = 1.25 inches, find thickness of 1 page

💰 Practical formulas

• Cost calculations: Registration fee + (monthly fee × number of months)
• Percent problems: Original price adjusted by successive discounts
• Break-even analysis: (Price per unit) × (number of units) = target amount

🎯 Problem-Solving Steps with Formulas

1️⃣ Understand the problem

• Identify what you're solving for
• Note what information is given
• Recognize which formula applies

2️⃣ Devise a plan—Translate

• Write down the relevant formula
• Express relationships between variables using the problem's constraints
• Example: "30% more on vacation than clothes" becomes an algebraic relationship

3️⃣ Carry out the plan—Solve

• Substitute known values into the equation
• Use algebraic techniques to isolate the unknown
• Perform calculations carefully (square roots, division, etc.)

4️⃣ Look—Check and interpret

• Verify the answer makes sense in context
• Substitute back into the original situation
• Check units and reasonableness

Don't confuse: The "check" step is not just re-doing the algebra—it's confirming the answer fits the real-world situation described in the problem.

🧮 Working with Constraints

🔗 Expressing one variable in terms of another

When a problem gives a relationship between variables:

• "Twice as long as wide" → length = 2 × width
• "30% more on vacation than clothes" → vacation = 1.3 × clothes
• "Height is 4 inches above sand level" → total height = sand height + 4 inches

This reduces a multi-variable problem to a single-variable equation you can solve.

📊 Multi-step formula problems

Some problems require:

• Applying a formula to find an intermediate value
• Using that result in a second calculation
• Example: Find area between two radii, then calculate storage density (GB per square cm)

⚠️ Unit awareness

• Convert units before substituting into formulas
• Example: Diameter must be converted to radius (divide by 2)
• Example: Speed in mph must be converted to meters/second for certain calculations

💡 Common Problem Types

🏊 Dimension problems

Given area/volume and one constraint, find dimensions:

• Rectangle with area and length-width relationship
• Cone with volume and given radius or height
• Sandbox with volume of sand and length-width dimensions

🕐 Rate problems

• Distance = speed × time
• Total cost = rate per unit × number of units
• Thickness = total thickness ÷ number of items

💵 Financial problems

• Total cost with fixed fees plus variable rates
• Discount problems with successive percentage reductions
• Break-even calculations

Example from excerpt: Shoes with 20% discount, then additional 10% off, final price $36—work backwards to find original price.