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ACT Math

ACT Math Help: Linear Equations

Review real example questions for Linear Equations in ACT Math.

Question 1

What is the value of xxx if 3x+6=153x + 6 = 153x+6=15?

  1. 5
  2. 4
  3. 2
  4. 3
Explanation: This is a two-step linear equation. To solve 3x + 6 = 15, first subtract 6 from both sides to get 3x = 9. Then divide both sides by 3 to isolate x: x = 3. The answer is 3.

Question 2

If 4y−7=2y+94y - 7 = 2y + 94y−7=2y+9, then y=?y = ?y=?

  1. 1
  2. 2
  3. 8
  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 3

Given K=12mv2K = \frac{1}{2}mv^2K=21​mv2, which expression equals vvv (for positive mmm, vvv, KKK)?

  1. 2Km\frac{2K}{m}m2K​
  2. K2m\sqrt{\frac{K}{2m}}2mK​​
  3. 2Km\sqrt{\frac{2K}{m}}m2K​​
  4. K2m\frac{\sqrt{K}}{2m}2mK​​
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 4

What is the solution to the equation 2x−3=x+72x - 3 = x + 72x−3=x+7?

  1. 5
  2. 12
  3. 8
  4. 10
Explanation: This is a linear equation with variable terms on both sides. To solve 2x - 3 = x + 7, subtract x from both sides: x - 3 = 7. Add 3 to both sides: x = 10. The solution is x = 10.

Question 5

What is the solution to the equation 2x+4=122x + 4 = 122x+4=12?

  1. 3
  2. 4
  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 6

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

  1. -13
  2. -5
  3. 13
  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 7

A store marks up an item after adding a fixed fee. If 3(x+4)=273(x + 4) = 273(x+4)=27, solve for xxx.

  1. 3
  2. 5
  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 8

A club collects dues. The total collected from xxx members is modeled by 2(x+7)=302(x + 7) = 302(x+7)=30. If 2(x+7)=302(x + 7) = 302(x+7)=30, what is xxx?​​

  1. 888
  2. 232323
  3. −8-8−8
  4. 161616
Explanation: This equation requires distributing first. To solve 2(x + 7) = 30, distribute the 2: 2x + 14 = 30. Subtract 14 from both sides: 2x = 16. Divide by 2: x = 8. Choice B (x = 23) likely forgot to distribute the 2.

Question 9

Two friends compare their savings. One has 3x+23x + 23x+2 dollars and the other has x+14x + 14x+14 dollars. If they have the same amount, solve for xxx in 3x+2=x+143x + 2 = x + 143x+2=x+14.​​

  1. 888
  2. 666
  3. −6-6−6
  4. 121212
Explanation: This equation has variables on both sides representing equal savings. To solve 3x + 2 = x + 14, subtract x from both sides: 2x + 2 = 14. Subtract 2: 2x = 12. Divide by 2: x = 6.

Question 10

What is the solution to the equation 2x+5=132x + 5 = 132x+5=13?

  1. 4
  2. 7
  3. 3
  4. 5
Explanation: This is a linear equation with a variable term and constant on one side. To solve 2x + 5 = 13, first subtract 5 from both sides: 2x = 8. Then divide both sides by 2: x = 4. The solution is x = 4.