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ACT Math Help: Linear Equations

Review real example questions for Linear Equations in ACT Math.

Question 1 / 10

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If $4y - 7 = 2y + 9$ , then $y = ?$

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Question 1

If $4y - 7 = 2y + 9$ , then $y = ?$

1
2
8 (correct answer)
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

Given $K = \frac{1}{2}mv^2$ , which expression equals $v$ (for positive $m$ , $v$ , $K$ )?

$\frac{2K}{m}$
$\sqrt{\frac{K}{2m}}$
$\sqrt{\frac{2K}{m}}$ (correct answer)
$\frac{\sqrt{K}}{2m}$

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

What is the solution to the equation $2x + 4 = 12$ ?

3
4 (correct answer)
5
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

A savings account balance changes by the same amount each week. If the relationship is $x - 9 = 4$ , solve for $x$ .

-13
-5
13 (correct answer)
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

A store marks up an item after adding a fixed fee. If $3(x + 4) = 27$ , solve for $x$ .

3
5 (correct answer)
9
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

A phone plan includes a base charge and an extra fee. If the total is modeled by $4x - 7 = 25$ , what is the value of $x$ ?

4
6
8 (correct answer)
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

If $3x - 7 = 4x + 2$ , then $x =$ ?

$-9$ (correct answer)
$-5$
$5$
$9$

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

What is the solution to the equation $4x + 7 = 2x - 9$ ?

$-1$
$8$
$-8$ (correct answer)
$1$

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

Solve for $x$ : $5 - 2x = 17$ .

$-6$ (correct answer)
$6$
$-12$
$12$

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

A temperature conversion is being simplified. If $5x + 2 = 3x + 18$ , what is the value of $x$ ?

6
8 (correct answer)
10
-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.