SAT Prep - Heart of Algebra (Linear Equations, Systems, Inequalities)

SAT – Heart of Algebra (Linear Equations, Systems, Inequalities)

SAT Heart of Algebra Study Guide

Topic: Linear Equations, Systems, Inequalities

What This Is

Heart of Algebra makes up ~30% of the SAT Math section (19–21 questions). It tests your ability to create, analyze, and solve linear equations, inequalities, and systems—skills critical for real-world problems like budgeting, data trends, or optimization. Example: "A gym charges a $50 membership fee plus $10 per class. Write an equation for the total cost (C) after x classes, then find how many classes you can attend with $200."

Key Terms & Rules

• Linear Equation in One Variable: An equation like 3x + 5 = 20; solve for x by isolating the variable.
• Slope-Intercept Form: y = mx + b; m = slope (rise/run), b = y-intercept (where the line crosses the y-axis).
• Point-Slope Form: y – y₁ = m(x – x₁); use when you know a point (x₁, y₁) and slope (m).
• Standard Form: Ax + By = C; A, B, and C are integers (no fractions/decimals). Slope = -A/B.
• Slope: m = (y₂ – y₁)/(x₂ – x₁); measures steepness. Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals (m₁ × m₂ = -1).
• System of Equations: Two or more equations with the same variables. Solutions are the (x, y) pair(s) that satisfy all equations.
• Substitution Method: Solve one equation for a variable, then plug into the other.
• Elimination Method: Add/subtract equations to cancel a variable (multiply first if needed).
• Linear Inequality: y > 2x + 1 or 3x – 5 ≤ 10. Graph with a dashed line (>) or solid line (≥), then shade above/below.
• Solution to a System of Inequalities: The overlapping shaded region on a graph.
• Word Problem Keywords:
• "Per" or "each" → slope (m).
• "Initial fee" or "starting amount" → y-intercept (b).
• "At least" → ≥; "no more than" → ≤.
• Calculator Shortcut: Use the "Y=" function to graph equations/inequalities and find intersections (2nd → TRACE → 5:intersect).

Step-by-Step / Process Flow

Solving a Linear Equation Word Problem

1. Read the question first → Underline what’s being asked (e.g., "How many classes can you attend?").
2. Define variables → Let x = number of classes, C = total cost.
3. Translate words to an equation → "$50 membership + $10 per class" → C = 10x + 50.
4. Plug in known values → Budget = $200 → 200 = 10x + 50.
5. Solve for x → Subtract 50: 150 = 10x → x = 15.
6. Check units/reasonableness → 15 classes × $10 = $150 + $50 = $200 ✔️.

Solving a System of Equations (Elimination)

1. Align equations →
   2x + 3y = 12
   4x – y = 5
2. Make coefficients opposites → Multiply the second equation by 3 to cancel y:
   12x – 3y = 15
3. Add equations → (2x + 3y) + (12x – 3y) = 12 + 15 → 14x = 27 → x = 27/14.
4. Substitute back → Plug x into 4x – y = 5 to find y.
5. Verify → Plug (x, y) into both original equations.

Common Mistakes

• Mistake: Forgetting to distribute a negative sign.
• Example: 3 – (2x + 5) = 10 → Incorrect: 3 – 2x + 5 = 10; Correct: 3 – 2x – 5 = 10.

• Correction: Always rewrite as 3 + (–1)(2x + 5) before distributing.

• Mistake: Mixing up slope and y-intercept in y = mx + b.

• Example: "A line has a slope of 2 and y-intercept of 3" → Incorrect: y = 3x + 2; Correct: y = 2x + 3.

• Correction: Remember "m" comes before "b" in the alphabet (and the equation).

• Mistake: Graphing inequalities with the wrong line/shading.

• Example: y ≥ 2x – 1 → Incorrect: Dashed line, shade below; Correct: Solid line, shade above.

• Correction: "Greater than" (>) or "greater than or equal to" (≥) → shade above; "less than" (<) or "less than or equal to" (≤) → shade below.

• Mistake: Solving for x but not answering the question.

• Example: "A taxi charges $3 per mile plus a $5 fee. How much does a 10-mile ride cost?" → Solve C = 3(10) + 5 = 35 but forget to write $35.

• Correction: Circle the question’s goal before solving.

• Mistake: Assuming parallel lines have the same y-intercept.

• Example: "Are y = 2x + 3 and y = 2x – 4 parallel?" → Incorrect: Yes, because slopes are equal and y-intercepts are different; Correct: Yes, because slopes are equal (y-intercepts don’t matter for parallelism).
• Correction: Parallel lines only need equal slopes.

Exam Insights

• Most-Tested Concepts:
• Slope from two points (e.g., "What’s the slope of the line through (1, 2) and (3, 8)?").
• Word problems (e.g., "A phone plan costs $20/month plus $0.10 per text. Write an equation for the total cost.").
• Systems of equations (especially elimination method).

• Graphing inequalities (e.g., "Which region satisfies y ≤ –x + 2 and y > 1/2x – 1?").

• Tricky Distinctions:

• "No solution" vs. "Infinite solutions":
  • No solution → Parallel lines (same slope, different y-intercepts; e.g., y = 2x + 1 and y = 2x – 3).
  • Infinite solutions → Same line (same slope and y-intercept; e.g., y = 2x + 1 and 2y = 4x + 2).

• "Or" vs. "And" in inequalities:

  
  
  • "x < 2 or x > 5" → Two separate regions on the number line.
  • "2 < x < 5" → One connected region.

• Common Distractors:

• Answer choices with reversed slope (e.g., y = 3x + 2 vs. y = 2x + 3).
• Incorrect inequality shading (e.g., shading above for y < 2x + 1).
• Misinterpreting "per" (e.g., "$5 per hour" → slope, not y-intercept).

Quick Check Questions

1. What is the slope of the line passing through (–2, 5) and (4, –1)?
2. A) –1
3. B) –1/2
4. C) 1

5. D) 2
   Answer: A) –1. Slope = (–1 – 5)/(4 – (–2)) = –6/6 = –1.

6. Solve the system:
   3x + 2y = 12
   x – y = 1

7. A) (2, 1)
8. B) (3, 2)
9. C) (4, 0)

10. D) (1, 2)
    Answer: A) (2, 1). Solve the second equation for x: x = y + 1. Substitute into the first equation: 3(y + 1) + 2y = 12 → 5y + 3 = 12 → y = 1. Then x = 2.

11. A bakery sells cupcakes for $2 each and cookies for $1 each. If Sarah buys 3 cupcakes and 4 cookies for $10, which system represents this?

12. A) 2c + k = 3 and 4c + k = 10
13. B) 2c + 4k = 3 and c + k = 10
14. C) 2c + k = 10 and 3c + 4k = 10
15. D) 3c + 4k = 10 and c = 2k
    Answer: C) 2c + k = 10 and 3c + 4k = 10. Let c = cupcakes, k = cookies. $2 per cupcake + $1 per cookie = $10 → 2c + k = 10. 3 cupcakes + 4 cookies = $10 → 3c + 4k = 10.

Last-Minute Cram Sheet

1. Slope formula: m = (y₂ – y₁)/(x₂ – x₁).
2. Slope-intercept form: y = mx + b (m = slope, b = y-intercept).
3. Parallel lines: Same slope; perpendicular lines: m₁ × m₂ = –1.
4. Elimination method: Multiply equations to cancel a variable.
5. Inequality shading: > or ≥ → shade above; < or ≤ → shade below.
6. ⚠️ Word problems: "Per" = slope; "initial" = y-intercept.
7. ⚠️ No solution = parallel lines (same slope, different y-intercepts).
8. ⚠️ Infinite solutions = same line (same slope and y-intercept).
9. Graphing shortcut: Use calculator’s "Y=" to find intersections.
10. ⚠️ Distribute negatives: –(2x + 3) = –2x – 3.