Lesson 5: SAT Math - Heart of Algebra & Linear Functions 📐

Introduction

Welcome to the Heart of Algebra, one of the three major content areas on SAT Math (alongside Problem-Solving & Data Analysis and Passport to Advanced Math). This domain accounts for approximately 33% of your math score—roughly 13 out of 58 questions! 🎯

The Heart of Algebra focuses on linear equations, inequalities, systems, and functions. These aren't just abstract math concepts—they model real-world situations like calculating costs, analyzing trends, and solving optimization problems. By the end of this lesson, you'll be able to tackle multi-step equations, interpret linear graphs, and solve systems with confidence.

What makes this lesson challenging? We're building on the problem-solving skills from Lesson 4 but adding algebraic abstraction. You'll need to translate word problems into equations, manipulate symbols fluently, and recognize when multiple solution methods exist.

💡 Pro Tip: The SAT loves testing algebra in context. Expect to see equations embedded in stories about business profits, distance-rate-time scenarios, and scientific relationships.

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Core Concept 1: Linear Equations & Manipulation 🔧

What Are Linear Equations?

A linear equation is an equation whose graph forms a straight line. The standard form is:

y = mx + b

Where:

• m = slope (rate of change)
• b = y-intercept (starting value)
• x = independent variable
• y = dependent variable

Key Manipulation Skills

The SAT expects you to rearrange equations into different forms:

1. Isolating Variables

Solve for x: 3x - 7 = 2x + 5

3x - 7 = 2x + 5
3x - 2x = 5 + 7    (subtract 2x, add 7)
x = 12

2. Dealing with Fractions

Solve: (2x + 3)/4 = 5

Multiply both sides by 4 first: 2x + 3 = 20, then x = 8.5

3. Equations with Variables on Both Sides

Solve: 5(x - 2) = 3x + 8

5x - 10 = 3x + 8    (distribute the 5)
5x - 3x = 8 + 10    (collect terms)
2x = 18
x = 9

💡 SAT Strategy: When you see answer choices with variables, consider plugging in the answers (backsolving) instead of solving algebraically. Start with choice C!

Understanding Slope 📊

Slope measures steepness and direction:

m = (y₂ - y₁)/(x₂ - x₁) = rise/run

• Positive slope: line rises left to right ↗️
• Negative slope: line falls left to right ↘️
• Zero slope: horizontal line ↔️
• Undefined slope: vertical line ↕️

🧠 Mnemonic: "Make Money Right Right" = m = rise/run (both Rs!)

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Core Concept 2: Systems of Linear Equations 🔗

A system of equations is two or more equations with the same variables. The SAT tests three solution methods:

Method 1: Substitution

Best when one equation is already solved for a variable.

Example: Solve the system:

y = 2x + 1
3x + y = 11

Substitute the first equation into the second:

3x + (2x + 1) = 11
5x + 1 = 11
5x = 10
x = 2

Then find y: y = 2(2) + 1 = 5 → Solution: (2, 5)

Method 2: Elimination

Best when coefficients line up nicely.

Example: Solve:

2x + 3y = 12
4x - 3y = 6

Add the equations (the 3y terms cancel):

  2x + 3y = 12
+ 4x - 3y = 6
_______________
  6x      = 18  →  x = 3

Substitute back: 2(3) + 3y = 12 → 3y = 6 → y = 2

Solution: (3, 2)

Method 3: Graphing

The solution is where the lines intersect. This method is useful when graphs are provided or you have a calculator.

        y
        |
      6 |    /
      5 |   •  (solution)
      4 |  /|
      3 | / |
      2 |/  |
      1 |   |
        |___|_____ x
        0  1 2 3 4

Types of Systems

+------------------+------------------+------------------+
|  One Solution    |  No Solution     | Infinite         |
|                  |                  | Solutions        |
+------------------+------------------+------------------+
| Lines intersect  | Parallel lines   | Same line        |
| at one point     | (never meet)     | (coincident)     |
+------------------+------------------+------------------+
| Different slopes | Same slope,      | Same slope AND   |
|                  | different        | same y-intercept |
|                  | y-intercepts     |                  |
+------------------+------------------+------------------+

🤔 Did you know? The SAT occasionally asks how many solutions a system has WITHOUT asking you to solve it. Check if the slopes and intercepts match!

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Core Concept 3: Linear Inequalities & Their Graphs 📉

Solving Inequalities

Inequalities work like equations with one critical exception:

⚠️ When you multiply or divide by a negative number, FLIP the inequality sign!

Solve: -3x + 5 > 14

-3x + 5 > 14
-3x > 9        (subtract 5)
x < -3         (divide by -3, FLIP the sign!)

Graphing Inequalities on a Number Line

      ◄─────────●═══════════►
    -5   -4   -3   -2   -1   0

• Open circle (○): < or > (value NOT included)
• Closed circle (●): ≤ or ≥ (value included)
• Shading: shows all solutions

Systems of Inequalities

When graphing on a coordinate plane:

1. Graph each inequality as if it were an equation
2. Use dashed lines for < or >, solid lines for ≤ or ≥
3. Shade the appropriate region
4. The solution region is where shaded areas overlap

Example: Graph y > 2x - 1 and y ≤ -x + 4

        y
      5 |     
      4 |   ●─────●
      3 |  /█████/
      2 | /█████/
      1 |/█████/
      0 |────────── x
       -1  0  1  2  3

The █ region represents solutions to BOTH inequalities.

💡 Test Point Method: After drawing boundary lines, pick a test point (like the origin 0,0) to determine which side to shade. Plug it into the inequality—if true, shade that side!

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Core Concept 4: Linear Functions & Modeling 🌍

Real-World Applications

Linear functions model situations with constant rates of change:

Cost Function: C(x) = mx + b

• m = variable cost per unit
• b = fixed cost
• x = number of units

Example Scenario: A gym charges a $50 enrollment fee plus $30 per month.

C(m) = 30m + 50

Where m = months of membership.

What does the gym cost for 8 months? C(8) = 30(8) + 50 = $290

Interpreting Function Notation

f(x) notation is just another way to write y:

• f(3) means "the value of f when x = 3"
• If f(x) = 2x - 5, then f(3) = 2(3) - 5 = 1

Rate of Change vs. Initial Value

+----------------------+------------------------+
| Rate of Change       | Initial Value          |
| (Slope, m)          | (y-intercept, b)       |
+----------------------+------------------------+
| "per hour"           | "starting amount"      |
| "each month"         | "initial fee"          |
| "every mile"         | "base charge"          |
| "for each ticket"    | "fixed cost"           |
+----------------------+------------------------+

🔧 Try this: A water tank contains 500 gallons and drains at 25 gallons per hour. Write the function: W(t) = -25t + 500 (note the negative rate!)

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Example 1: Multi-Step Equation with Distribution 🎯

Problem: If 3(2x - 5) = 4x + 7, what is the value of x?

Solution:

Step 1: Distribute the 3

6x - 15 = 4x + 7

Step 2: Collect x terms on one side

6x - 4x = 7 + 15
2x = 22

Step 3: Solve

x = 11

Why this matters: The SAT often embeds this type of equation in word problems. You might see "Three times the difference of twice a number and five equals..." You need to translate that into 3(2x - 5).

💡 SAT Shortcut: Before distributing, check if you can factor instead. Sometimes the algebra simplifies faster!

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Example 2: System of Equations in Context 🛒

Problem: At a farmer's market, apples cost $2 per pound and oranges cost $3 per pound. Maria buys 12 pounds of fruit for $31. How many pounds of apples did she buy?

Solution:

Step 1: Define variables

• Let a = pounds of apples
• Let o = pounds of oranges

Step 2: Write equations

a + o = 12       (total pounds)
2a + 3o = 31     (total cost)

Step 3: Solve by substitution From equation 1: o = 12 - a

Substitute into equation 2:

2a + 3(12 - a) = 31
2a + 36 - 3a = 31
-a = -5
a = 5

Answer: Maria bought 5 pounds of apples.

Why this matters: The SAT loves mixture problems (fruit, tickets, coins). Always start by clearly defining your variables and identifying what each equation represents.

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Example 3: Inequality with Real-World Constraint 🚗

Problem: A rental car company charges $40 per day plus $0.25 per mile. If Sarah has a budget of $100, what is the maximum number of miles she can drive on a one-day rental?

Solution:

Step 1: Write an inequality

40 + 0.25m ≤ 100

Where m = miles driven

Step 2: Solve

0.25m ≤ 60
m ≤ 240

Answer: Sarah can drive up to 240 miles.

Why this matters: Budget constraints are classic SAT scenarios. The key word "maximum" signals you need an inequality, not an equation. Watch for "at most" (≤), "at least" (≥), "more than" (>), and "less than" (<).

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Example 4: Parallel and Perpendicular Lines 🛤️

Problem: Line ℓ passes through points (2, 5) and (6, 13). What is the slope of a line perpendicular to line ℓ?

Solution:

Step 1: Find the slope of line ℓ

m = (y₂ - y₁)/(x₂ - x₁)
m = (13 - 5)/(6 - 2)
m = 8/4 = 2

Step 2: Apply the perpendicular slope rule Perpendicular slopes are negative reciprocals: flip and change sign.

Slope of ℓ: 2/1
Perpendicular slope: -1/2

Answer: The perpendicular slope is -1/2.

🧠 Memory trick: Perpendicular slopes multiply to -1. Check: 2 × (-1/2) = -1 ✓

Parallel lines have the SAME slope (no reciprocal, no sign change).

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Common Mistakes to Avoid ⚠️

Mistake 1: Forgetting to Flip the Inequality Sign

❌ Wrong: -2x > 6 → x > -3 ✅ Correct: -2x > 6 → x < -3 (flipped!)

Prevention: Circle negative coefficients when solving inequalities as a visual reminder.

Mistake 2: Confusing Parallel vs. Perpendicular Slopes

❌ Wrong: "A line with slope 3 is perpendicular to a line with slope 1/3" ✅ Correct: It's perpendicular to slope -1/3 (must be negative!)

Prevention: Remember: perpendicular = negative reciprocal (both operations required).

Mistake 3: Distribution Errors

❌ Wrong: 3(x + 2) = 3x + 2 ✅ Correct: 3(x + 2) = 3x + 6

Prevention: Distribute to EVERY term inside the parentheses. Draw arrows to each term.

Mistake 4: Losing Track of Variables in Word Problems

❌ Wrong: Defining x as "apples" then writing an equation using "pounds of apples" ✅ Correct: Be specific: "Let x = number of pounds of apples"

Prevention: Write your variable definitions on your scratch paper before starting equations.

Mistake 5: Solving for the Wrong Variable

❌ The question asks for y, but you solve for x and stop. ✅ After finding x, substitute back to find y.

Prevention: Circle what the question is asking for. Double-check before bubbling your answer.

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Key Takeaways 🎓

✅ Linear equations follow the form y = mx + b, where m is slope and b is y-intercept

✅ Solving systems: Use substitution (when one variable is isolated), elimination (when coefficients align), or graphing

✅ Inequalities: Flip the sign when multiplying/dividing by negatives; use open circles for < or >, closed for ≤ or ≥

✅ Perpendicular slopes are negative reciprocals; parallel slopes are identical

✅ Word problems: Define variables clearly, identify what each equation represents, and always check what the question asks for

✅ Function notation: f(3) means "plug in 3 for x"; the slope represents rate of change, the y-intercept represents initial value

✅ SAT strategies: Plug in answer choices for equation questions, use test points for inequality shading, check units in word problems

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📋 Quick Reference Card

╔════════════════════════════════════════════════════════╗
║         SAT HEART OF ALGEBRA CHEAT SHEET             ║
╠════════════════════════════════════════════════════════╣
║ SLOPE-INTERCEPT FORM:  y = mx + b                    ║
║                        m = slope, b = y-intercept     ║
║                                                        ║
║ SLOPE FORMULA:  m = (y₂-y₁)/(x₂-x₁) = rise/run      ║
║                                                        ║
║ PARALLEL LINES:        Same slope                     ║
║ PERPENDICULAR LINES:   Negative reciprocal slopes     ║
║                        (m₁ × m₂ = -1)                 ║
║                                                        ║
║ SYSTEMS - SUBSTITUTION:                               ║
║   1. Solve one equation for one variable              ║
║   2. Substitute into the other equation               ║
║   3. Solve, then back-substitute                      ║
║                                                        ║
║ SYSTEMS - ELIMINATION:                                ║
║   1. Align equations vertically                       ║
║   2. Add/subtract to eliminate a variable             ║
║   3. Solve, then back-substitute                      ║
║                                                        ║
║ INEQUALITIES:                                         ║
║   • Solve like equations BUT...                       ║
║   • FLIP sign when multiplying/dividing by negative   ║
║   • Graph: ○ for < or >, ● for ≤ or ≥               ║
║                                                        ║
║ WORD PROBLEM KEYWORDS:                                ║
║   • "per/each/every" → rate of change (slope)        ║
║   • "initial/starting/fixed" → y-intercept           ║
║   • "at most/maximum" → ≤                            ║
║   • "at least/minimum" → ≥                           ║
╚════════════════════════════════════════════════════════╝

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📚 Further Study

1. Khan Academy SAT Math - Heart of Algebra: https://www.khanacademy.org/sat/heart-of-algebra - Comprehensive video lessons with practice exercises aligned to SAT format

2. College Board Official SAT Practice: https://collegereadiness.collegeboard.org/sat/practice/full-length-practice-tests - Full-length practice tests with detailed answer explanations showing algebraic techniques

3. Desmos Graphing Calculator Tutorial: https://www.desmos.com/calculator - Interactive graphing tool perfect for visualizing linear functions and systems (permitted on SAT digital test)

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🎯 You're now halfway through your SAT Math preparation! The Heart of Algebra forms the foundation for more advanced topics. In our next lesson, we'll tackle Passport to Advanced Math, including quadratics, exponentials, and polynomial functions. Keep practicing these linear skills—they appear in nearly every SAT Math section!

💪 Practice tip: Time yourself on 5-6 algebra problems in one sitting. Aim for 1-1.5 minutes per question to build test-day speed.