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 7a + 2b = 1 and 3a + 2b = −11, what is the value of b?
- A.
  −10
- B.
  3
- C.
  9.25
- D.
  10
Answer: A. −10
Subtract: 4a = 1 − (−11) = 12, so a = 3, and then 2b = 1 − 21 = −20, so b = −10. 3 is a, not b; 9.25 computes 1 − 11 instead of 1 − (−11), giving a = −2.5; 10 drops the sign.
Question 2
Priya is 5 years older than twice her brother's age, and the sum of their ages is 47. How old is Priya?
- A.
  14
- B.
  23.5
- C.
  28
- D.
  33
Answer: D. 33
With p = 2b + 5 and p + b = 47, substitution gives 3b + 5 = 47, so b = 14 and p = 33. 14 is the brother's age; 23.5 splits 47 evenly; 28 is twice the brother's age without the extra 5.
Question 3
A school in Dubai bought 55 water-park tickets over two visits: weekday tickets at AED 180 and weekend tickets at AED 240. It spent AED 2,700 more on weekend tickets than on weekday tickets. How many weekend tickets did it buy?
- A.
  25
- B.
  27.5
- C.
  30
- D.
  45
Answer: C. 30
With w + d = 55 and 240w − 180d = 2,700: 240w − 180(55 − w) = 2,700 gives 420w = 12,600, so w = 30 (and d = 25). 25 is the weekday count; 27.5 splits 55 evenly; 45 divides 2,700 by the price difference, 60.
Question 4
In the standard (x, y) coordinate plane, the graphs of y = x² + 1 and y = 3x + 5 intersect at two points. What is the sum of the y-coordinates of the two points?
- A.
  3
- B.
  14
- C.
  17
- D.
  19
Answer: D. 19
x² + 1 = 3x + 5 gives x² − 3x − 4 = 0, so x = 4 or x = −1, and the points are (4, 17) and (−1, 2). The sum of the y-coordinates is 19. 3 is the sum of the x-coordinates; 17 is one y-value; 14 computes 3(3) + 5, adding the 5 only once.
Question 5
If (x + 1)/3 + y/2 = 5 and x/4 − (y − 2)/8 = 3/4, what is the value of x?
- A.
  5
- B.
  6
- C.
  6.5
- D.
  11
Answer: A. 5
Multiply by 6: 2x + 2 + 3y = 30, so 2x + 3y = 28. Multiply by 8: 2x − (y − 2) = 6, so 2x − y = 4. Subtracting gives 4y = 24, y = 6 and x = 5. 6 is y; 6.5 writes −(y − 2) as −y − 2; 11 is x + y.

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