ACT · Math · Preparing for Higher Math: Algebra

Linear equations in two variables

Linear equations in two variables 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 points on a two-variable line, two-variable model from a context, interpret a coefficient or constant and convert between forms.

Common traps: swapping x and y; substituting into only one term; a sign error with a negative.

Sample questions

Question 1
A robot drives along the line 3x + 4y = 24 on a coordinate grid measured in meters. It starts where the line meets the y-axis and stops where it meets the x-axis. How many meters does it travel?
- A.
  6
- B.
  8
- C.
  10
- D.
  14
Answer: C. 10
The intercepts are (0, 6) and (8, 0). The distance is √(8² + 6²) = 10 m. Choice A (6) gives the y-intercept only. Choice B (8) gives the x-intercept only. Choice D (14) adds the two intercepts.
Question 2
A family drives from Houston to San Antonio in a car that gets 30 miles per gallon. Gasoline costs $3.60 per gallon, and the family has already paid a $12 toll. Which equation gives the total cost C, in dollars, of the toll and fuel after driving d miles?
- A.
  C = 12d + 0.12
- B.
  C = 0.12d + 12
- C.
  C = 8.33d + 12
- D.
  C = 108d + 12
Answer: B. C = 0.12d + 12
Fuel costs 3.60 ÷ 30 = $0.12 per mile, and the toll is a fixed $12, so C = 0.12d + 12. Choice A (C = 12d + 0.12) swaps the rate and the starting amount. Choice C (C = 8.33d + 12) divides 30 by 3.6 instead of 3.6 by 30. Choice D (C = 108d + 12) multiplies 30 by 3.6.
Question 3
The line y = mx − 7 passes through the point (4, 5) in the standard (x, y) coordinate plane. What is the value of m?
- A.
  −3
- B.
  −1/2
- C.
  11/5
- D.
  3
Answer: D. 3
5 = 4m − 7 gives 4m = 12, so m = 3. −3 slips the sign; −1/2 subtracts 7 instead of adding it (4m = −2); 11/5 substitutes the point as (5, 4), giving 4 = 5m − 7.
Question 4
In the standard (x, y) coordinate plane, the lines 6x − ky = 3 and 2x + 5y = 9 are parallel. What is the value of k?
- A.
  −15
- B.
  −5
- C.
  2.4
- D.
  15
Answer: A. −15
Slopes: 6/k for the first line and −2/5 for the second. Setting 6/k = −2/5 gives k = −15. 15 drops the sign; −5 copies the y-coefficient without scaling by 3; 2.4 is the value that makes the lines perpendicular.
Question 5
A gift shop in Muscat sells bracelets for OMR 12 and key rings for OMR 8. Zainab spends exactly OMR 100 and buys at least one of each. What is the greatest number of bracelets she can buy?
- A.
  2
- B.
  7
- C.
  8
- D.
  11
Answer: B. 7
12b + 8r = 100 simplifies to 3b + 2r = 25, so 3b must be odd and b must be odd. b = 7 leaves 25 − 21 = 4, so r = 2 (b = 8 would need half a key ring). 8 is 100 ÷ 12 rounded down, ignoring whole key rings; 2 is the key-ring count; 11 is the most key rings possible.

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