ACT Math Quiz: Linear Equations

What this quiz covers

This quiz focuses on Linear Equations, giving you a quick way to practice the rules, question types, and explanations that matter most for ACT Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. 1
2. 2
3. 8 (correct answer)
4. 16

Explanation: The correct answer is C (8). Solve the linear equation: 4y − 7 = 2y + 9. Subtract 2y from both sides: 2y − 7 = 9. Add 7 to both sides: 2y = 16. Divide by 2: y = 8. D (16) is the most common error — students correctly reach 2y = 16 but forget to divide by 2, reporting 16 as the answer. B (2) results from dividing 16 by 8 (the answer) instead of by 2 — a circular error. A (1) comes from a more serious arithmetic mistake earlier in the process. Always check your answer by substituting back: 4(8) − 7 = 25 = 2(8) + 9 ✓.

Question 2

1. $\frac{2K}{m}$
2. $\sqrt{\frac{K}{2m}}$
3. $\sqrt{\frac{2K}{m}}$ (correct answer)
4. $\frac{\sqrt{K}}{2m}$

Explanation: This is a literal equations question testing multi-step algebraic isolation. Choice C (√(2K/m)) is correct — isolate v: K = (1/2)mv² → 2K = mv² → v² = 2K/m → v = √(2K/m). Choice A (2K/m) correctly isolates v² but forgets to take the square root — stopping one step early. Choice B (√(K/2m)) divides K by 2m instead of multiplying: student moves the (1/2) to the denominator rather than multiplying both sides by 2, giving v² = K/(m/2)... actually that gives K/(m/2) = 2K/m, which is correct. B may come from: K = (1/2)mv² → K/m = (1/2)v² → v² = K/(m/2) → student writes √(K/2m) incorrectly. Most likely B = student writes v = √(K/2m) from not doubling K. Choice D (√K / 2m) takes the square root of K alone without dividing by m. Pro tip: When isolating a squared variable, handle all multiplication/division first, then take the square root last. Write each step: 2K = mv² (multiply both sides by 2), then v² = 2K/m (divide both sides by m), then v = √(2K/m) (square root both sides).

Question 3

1. 3
2. 4 (correct answer)
3. 5
4. 6

Explanation: This is a two-step linear equation. To solve 2x + 4 = 12, first subtract 4 from both sides to get 2x = 8. Then divide both sides by 2 to isolate x: x = 4. The answer is 4.

Question 4

1. -13
2. -5
3. 13 (correct answer)
4. 5

Explanation: This is a basic linear equation x - 9 = 4 representing a savings account change, with x as the balance or related value. Add 9 to both sides to isolate x: x = 4 + 9. This simplifies to x = 13. Therefore, the value of x is 13, which is choice C. Choice D (5) could be from subtracting 9 instead of adding, getting 4 - 9 = -5, but that's choice B; careful sign handling is key.

Question 5

1. 3
2. 5 (correct answer)
3. 9
4. 13

Explanation: This is a linear equation with distribution: 3(x + 4) = 27 marks up an item with a fee, where x is the base value. Distribute the 3: 3x + 12 = 27. Subtract 12 from both sides: 3x = 15. Divide by 3: x = 5. Therefore, the value of x is 5, which is choice B. Choice C (9) might result from dividing 27 by 3 without distributing, getting x + 4 = 9 and forgetting to subtract 4.

Question 6

1. 4
2. 6
3. 8 (correct answer)
4. 9

Explanation: This is a linear equation 4x - 7 = 25 modeling a phone plan total with base and extra fees, where x is the variable. Add 7 to both sides: 4x = 32. Divide both sides by 4: x = 8. Therefore, the value of x is 8, matching choice C. Choice B (6) might occur from subtracting 7 instead of adding, leading to 4x = 18 and x=4.5, but that's not a choice; precision in operations is essential.

Question 7

1. $-9$ (correct answer)
2. $-5$
3. $5$
4. $9$

Explanation: This is a linear equations question testing one-variable solving. Choice A (−9) is correct — subtract 4x from both sides: −x − 7 = 2. Add 7: −x = 9. Divide both sides by −1: x = −9. Choice B (−5) results from a subtraction error when combining terms, arriving at −x = 5 instead of −x = 9. Choice C (5) solves −x = −5 (a sign error earlier in the process), giving x = 5. Choice D (9) correctly arrives at −x = 9 but then forgets to divide by −1, reporting x = 9 instead of x = −9. Pro tip: When you reach the step −x = [number], always divide both sides by −1 and flip the sign. The equation is NOT solved until x (not −x) is isolated.

Question 8

1. $-1$
2. $8$
3. $-8$ (correct answer)
4. $1$

Explanation: This equation has variables on both sides. Starting with 4x + 7 = 2x - 9, subtract 2x from both sides: 2x + 7 = -9. Subtract 7 from both sides: 2x = -16. Divide by 2: x = -8.

Question 9

1. $-6$ (correct answer)
2. $6$
3. $-12$
4. $12$

Explanation: This equation has a negative coefficient for x. Starting with 5 - 2x = 17, subtract 5 from both sides: -2x = 12. Divide by -2: x = -6. Remember that dividing by a negative number changes the sign.

Question 10

1. 6
2. 8 (correct answer)
3. 10
4. -8

Explanation: This is a linear equation 5x + 2 = 3x + 18 simplifying a temperature conversion, where x is the variable to solve. Subtract 3x from both sides: 2x + 2 = 18. Subtract 2 from both sides: 2x = 16. Divide by 2: x = 8. Therefore, the value of x is 8, matching choice B. Choice C (10) might result from subtracting 2x instead, leading to arithmetic errors.

Question 11

1. $\tfrac{11}{5}$
2. $5$ (correct answer)
3. $\tfrac{25}{5}$
4. $-5$

Explanation: This is a linear equation that we solve by isolating x. Starting with $5x - 7 = 18$, add 7 to both sides to get $5x = 25$. Divide both sides by 5 to get $x = 5$. The value of x is $5$.

Question 12

1. $6$
2. $8$ (correct answer)
3. $-8$
4. $16$

Explanation: This is a linear equation with variables on both sides. Starting with x + 14 = 3x - 2, subtract x from both sides to get 14 = 2x - 2. Add 2 to both sides to get 16 = 2x. Divide by 2 to get x = 8.

Question 13

1. 10 (correct answer)
2. 15
3. 5
4. 8

Explanation: This is a linear equation with parentheses. To solve 2(x - 5) = 10, first distribute the 2: 2x - 10 = 10. Add 10 to both sides: 2x = 20. Divide by 2: x = 10. The answer is 10.

Question 14

1. 3
2. 4 (correct answer)
3. 5
4. 6

Explanation: This is a linear equation with parentheses. To solve $3(x + 4) = 24$, first distribute the 3: $3x + 12 = 24$. Subtract 12 from both sides: $3x = 12$. Divide by 3: $x = 4$. The answer is 4.

Question 15

1. $-8$
2. $30$
3. $8$ (correct answer)
4. $19$

Explanation: This is a linear equation with variables on both sides. Starting with 2x + 11 = x + 19, subtract x from both sides to get x + 11 = 19. Subtract 11 from both sides to get x = 8. The solution is x = 8.

Question 16

1. $\tfrac{28}{3}$
2. $6$
3. $-4$
4. $4$ (correct answer)

Explanation: This is a linear equation where we need to isolate the variable x. Starting with 3x + 8 = 20, subtract 8 from both sides to get 3x = 12. Divide both sides by 3 to get x = 4. The value of x is 4.

Question 17

1. 4
2. 5
3. 7
4. 6 (correct answer)

Explanation: This is a linear equation with variable terms on both sides. To solve 12x + 4 = 8x + 28, subtract 8x from both sides: 4x + 4 = 28. Subtract 4 from both sides: 4x = 24. Divide by 4: x = 6. The solution is x = 6.

Question 18

1. 4
2. 5 (correct answer)
3. 6
4. 8

Explanation: This is a linear equation with distribution: $5(x + 1) - 10 = 20$ models a bill split with discount, where x is the variable. Distribute the 5: $5x + 5 - 10 = 20$. Combine like terms: $5x - 5 = 20$. Add 5 to both sides: $5x = 25$. Divide by 5: $x = 5$. Therefore, the value of x is 5, matching choice B. Choice C (6) might occur from forgetting to subtract 10 properly, leading to $5x + 5 = 20$ and errors.

Question 19

1. 7
2. 8
3. 9 (correct answer)
4. 10

Explanation: This is a simple linear equation 6x = 54 representing total items in boxes, with x as items per box. Divide both sides by 6: x = 54 / 6. This simplifies to x = 9. Therefore, the value of x is 9, matching choice C. Choice B (8) might come from misdividing 48 by 6 or subtracting before dividing.

Question 20

1. 6
2. 4
3. 3 (correct answer)
4. 5

Explanation: This is a two-step linear equation. To solve 6x + 2 = 20, first subtract 2 from both sides to get 6x = 18. Then divide both sides by 6 to isolate x: x = 3. The answer is 3.