ACT · Math · Preparing for Higher Math: Algebra

Linear equations

Linear equations is part of Preparing for Higher Math: Algebra in the ACT Math section. The questions below are written in the style of the real test, each with the answer and a worked explanation.

Question types you will see here include multi-step linear equation, equation from a fixed-fee-plus-rate context, solve a formula for one variable and find an expression's value without solving.

Common traps: sign error when distributing a negative; combining unlike terms; dividing only one term by the coefficient.

Sample questions

Question 1
If 3x + 13 = 4x + 17, what is the value of x?
- A.
  -30
- B.
  -4
- C.
  4/7
- D.
  4
Answer: B. -4
Subtract 4x from both sides to get -x + 13 = 17. Subtract 13 from both sides: -1x = 4. Divide by -1: x = -4.
Question 2
A moving company charges $25 for the truck plus $12 per box. If Hana paid a total of $301, how many boxes were paid for?
- A.
  23
- B.
  24
- C.
  25
- D.
  27
Answer: A. 23
Let n be the number of boxes. Then 25 + 12n = 301. Subtract 25: 12n = 276. Divide by 12: n = 23.
Question 3
If 6x + 1 = 8x - 19, what is the value of x?
- A.
  -10
- B.
  -10/7
- C.
  9
- D.
  10
Answer: D. 10
Subtract 8x from both sides to get -2x + 1 = -19. Subtract 1 from both sides: -2x = -20. Divide by -2: x = 10.
Question 4
In a mobile game, the number n of a certain level has this property: 3 more than twice n is equal to 5 less than 4 times n. What is the value of n?
- A.
  −1
- B.
  4
- C.
  5.5
- D.
  11.5
Answer: B. 4
Translate: 2n + 3 = 4n − 5. Then 8 = 2n, so n = 4. Check: 2(4) + 3 = 11 and 4(4) − 5 = 11. Choice A (−1) reads 5 less than as 5 more (2n + 3 = 4n + 5). Choice C (5.5) doubles the 3 as well (2(n + 3) = 4n − 5). Choice D (11.5) takes 5 less than as 4(n − 5).
Question 5
A car-rental company in Cape Town offers two one-day plans. Plan A costs 350 rand plus 2.50 rand per kilometer driven, and Plan B costs 500 rand plus 1.25 rand per kilometer driven. For how many kilometers driven in one day do the two plans cost the same?
- A.
  120
- B.
  150
- C.
  340
- D.
  375
Answer: A. 120
Set the costs equal: 350 + 2.50k = 500 + 1.25k, so 1.25k = 150 and k = 120 km (both plans then cost 650 rand). Choice B stops at the difference in the daily fees, 150, without dividing by the difference in the per-kilometer rates; choice C divides the sum of the daily fees, 850, by 2.50; choice D multiplies 150 by 2.50 instead of dividing by 1.25.

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