ACT · Math · Preparing for Higher Math: Algebra

Systems of equations

Systems of 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 solve by elimination, solve by substitution, two-unknown word problem and number of solutions.

Common traps: adding when subtraction is needed; multiplying only one side of an equation; reporting x when y was asked.

Sample questions

Question 1
If 3x + y = 17 and x + 3y = 11, what is the value of x + y?
- A.
  3
- B.
  7
- C.
  14
- D.
  28
Answer: B. 7
Adding the equations gives 4x + 4y = 28, so x + y = 7 without solving for each variable. 28 stops before dividing by 4; 14 divides by 2 instead of 4; 3 subtracts the equations, which gives x − y.
Question 2
In the standard (x, y) coordinate plane, line ℓ passes through (0, 7) and (7, 0), and line m passes through (0, −2) and (1, 0). What are the coordinates of the point where ℓ and m intersect?
- A.
  (0, 7)
- B.
  (1, 0)
- C.
  (3, 4)
- D.
  (4, 3)
Answer: C. (3, 4)
ℓ is y = −x + 7 and m is y = 2x − 2. Setting −x + 7 = 2x − 2 gives x = 3, and y = 4. A is only ℓ's y-intercept and B only m's x-intercept; neither lies on both lines. D reverses the coordinates.
Question 3
A cricket analyst models two quantities with y = 3x − 5 and 2x + y = 15. What is the value of x?
- A.
  4
- B.
  7
- C.
  11
- D.
  20
Answer: A. 4
Substitute: 2x + (3x − 5) = 15, so 5x = 20 and x = 4 (then y = 7). Choice B (7) gives y instead of x. Choice C (11) gives x + y. Choice D (20) forgets to divide 5x = 20 by 5.
Question 4
A lab technician in Hyderabad mixes a 20% saline solution with a 50% saline solution to make 30 litres of a 40% solution. How many litres of the 20% solution does she use?
- A.
  10
- B.
  15
- C.
  20
- D.
  24
Answer: A. 10
With x + y = 30 and 0.2x + 0.5y = 0.4(30) = 12: 0.2x + 15 − 0.5x = 12 gives x = 10 (and y = 20). 15 splits the volume evenly, as if averaging the strengths; 20 is the 50% amount; 24 divides 12 by 0.5.
Question 5
For what value of k does the system of equations below have infinitely many solutions?
4x + 6y = 15 10x + ky = 37.5
- A.
  6
- B.
  10
- C.
  15
- D.
  25
Answer: C. 15
The second equation must be 2.5 times the first: 2.5 × 4 = 10, 2.5 × 6 = 15, and 2.5 × 15 = 37.5. So k = 15, and the lines coincide. Choice A (6) copies the y-coefficient of the first equation. Choice B (10) copies the x-coefficient of the second equation. Choice D (25) multiplies 10 by the scale factor 2.5 instead of 6.

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