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ACT Math · Learn by Concept

ACT Math Help: Systems Of Equations

Review real example questions for Systems Of Equations in ACT Math.

Question 1 / 10

0 of 10 answered

What is the xxx-value of the solution to the system: 3x+y=143x + y = 143x+y=14 and x−y=2x - y = 2x−y=2?

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Question 1

What is the xxx-value of the solution to the system: 3x+y=143x + y = 143x+y=14 and x−y=2x - y = 2x−y=2?

  1. 2
  2. 4 (correct answer)
  3. 8
  4. 12

Explanation: This is a systems of equations question testing the elimination method. Choice B (4) is correct — add the two equations to eliminate y: (3x + y) + (x − y) = 14 + 2 → 4x = 16 → x = 4. (If desired, verify: y = x − 2 = 2, and 3(4) + 2 = 14 ✓.) Choice A (2) is the y-value, not x — the student may have solved for y and stopped, or substituted back into the wrong equation. Choice C (8) comes from solving 4x = 16 as x = 16/2 = 8, dividing by 2 instead of 4. Choice D (12) likely comes from adding the right-hand sides to get 16, then adding 14 − 2 = 12 via some misalignment. Pro tip: When one variable has matching coefficients with opposite signs (y and −y here), adding the equations eliminates that variable immediately. Always double-check by substituting both values back into BOTH original equations.

Question 2

Tickets to a high school play cost 12foradultsand12 for adults and 12foradultsand8 for children. On opening night, 150 tickets were sold and total revenue was $1,440. How many more children's tickets were sold than adult tickets?

  1. 303030 (correct answer)
  2. 454545
  3. 606060
  4. 909090

Explanation: This is a systems of equations word problem testing multi-step algebraic modeling. Choice A (30) is correct — set up: let A = adult tickets and C = children's tickets. Two equations: A + C = 150 (total tickets) and 12A + 8C = 1,440 (total revenue). Solve by substitution: C = 150 − A → 12A + 8(150 − A) = 1,440 → 12A + 1,200 − 8A = 1,440 → 4A = 240 → A = 60. Then C = 150 − 60 = 90. Difference: C − A = 90 − 60 = 30. Choice B (45) likely comes from an arithmetic error mid-solve, perhaps computing 4A = 180 → A = 45. Choice C (60) reports the number of adult tickets — finding A but not completing the final step (finding the difference). Choice D (90) reports the number of children's tickets — finding C but not subtracting A. Pro tip: Systems word problems require a final step after solving for variables. Read the question again carefully — here it asks "how many MORE children's tickets," which means C − A, not just C or A alone. After solving the system, always return to the original question to make sure you're reporting the right quantity.

Question 3

A taxi charges a \3flatfeeplusflat fee plusflatfeeplus$2permile.Anothertaxichargesaper mile. Another taxi charges apermile.Anothertaxichargesa$1flatfeeplusflat fee plusflatfeeplus$3permile.Forwhatnumberofmilesper mile. For what number of milespermile.Forwhatnumberofmilesx$ do the two taxis cost the same (in dollars)?

y = 2x + 3\\ y = 3x + 1 \end{cases}$$
  1. 111
  2. 222 (correct answer)
  3. 333
  4. 444

Explanation: Set the costs equal to find when taxis charge the same. We have 2x + 3 = 3x + 1. Subtract 2x from both sides: 3 = x + 1. Subtract 1 from both sides: x = 2. At 2 miles, both taxis cost $7.

Question 4

If 4x+y=104x + y = 104x+y=10 and 2x−y=22x - y = 22x−y=2, what is the value of xxx?

  1. 1
  2. 2 (correct answer)
  3. 3
  4. 4

Explanation: Use elimination to solve this system. Add the two equations: (4x + y) + (2x - y) = 10 + 2, which gives 6x = 12, so x = 2. This matches choice B.

Question 5

If 2x+3y=132x + 3y = 132x+3y=13 and 2x−3y=12x - 3y = 12x−3y=1, what is the value of yyy?

  1. 1
  2. 2 (correct answer)
  3. 3
  4. 4

Explanation: Use the elimination method by adding 2x+3y=132x + 3y = 132x+3y=13 and 2x−3y=12x - 3y = 12x−3y=1. This eliminates y, giving 4x=144x = 144x=14, so x=14/4=3.5x = 14/4 = 3.5x=14/4=3.5. Substitute into 2x−3y=12x - 3y = 12x−3y=1: 2(3.5)−3y=12(3.5) - 3y = 12(3.5)−3y=1, 7−3y=17 - 3y = 17−3y=1, −3y=−6-3y = -6−3y=−6, y=2y = 2y=2. Alternatively, subtract the equations to get 6y=126y = 126y=12, y=2y = 2y=2 directly. Choice C of 3 might come from misadding to 4x=124x = 124x=12.

Question 6

Solve the system of equations:

5x - y = 14\\ 2x + y = 7 \end{cases}$$ What is the solution $(x, y)$?
  1. (3,1)(3, 1)(3,1) (correct answer)
  2. (2,3)(2, 3)(2,3)
  3. (1,3)(1, 3)(1,3)
  4. (3,−1)(3, -1)(3,−1)

Explanation: Use elimination by adding the equations directly. Adding 5x - y = 14 and 2x + y = 7 gives 7x = 21, so x = 3. Substitute x = 3 into 2x + y = 7: 2(3) + y = 7, so y = 1. The solution is (3, 1).

Question 7

Two numbers have sum 13 and difference 5. If x+y=13x+y=13x+y=13 and x−y=5x-y=5x−y=5, what is (x,y)(x,y)(x,y)?

  1. (7,6)(7,6)(7,6)
  2. (4,9)(4,9)(4,9)
  3. (8,5)(8,5)(8,5)
  4. (9,4)(9,4)(9,4) (correct answer)

Explanation: Use elimination to solve this system with x+y=13x + y = 13x+y=13 and x−y=5x - y = 5x−y=5. Adding the equations eliminates yyy: 2x=182x = 182x=18, so x=9x = 9x=9. Substitute back: 9+y=139 + y = 139+y=13, so y=4y = 4y=4. The solution is (9,4)(9, 4)(9,4).

Question 8

What is the solution (x,y)(x, y)(x,y) to the system x+y=9x + y = 9x+y=9 and x−y=1x - y = 1x−y=1?

  1. (7, 2)
  2. (6, 3)
  3. (4, 5)
  4. (5, 4) (correct answer)

Explanation: Use elimination to solve this system. Adding x + y = 9 and x - y = 1 eliminates y: 2x = 10, so x = 5. Substituting into x + y = 9: 5 + y = 9, so y = 4. The solution is (5, 4).

Question 9

If the system below is true, what is the value of yyy?

3x + 4y = 1\\ 6x - 4y = 11 \end{cases}$$
  1. −54-\tfrac{5}{4}−45​
  2. −34-\tfrac{3}{4}−43​ (correct answer)
  3. 54\tfrac{5}{4}45​
  4. 34\tfrac{3}{4}43​

Explanation: Use elimination by adding the equations. Adding 3x + 4y = 1 and 6x - 4y = 11 gives 9x = 12, so x = 4/3. Substitute into 3x + 4y = 1: 3(4/3) + 4y = 1, which gives 4 + 4y = 1, so 4y = -3 and y = -3/4.

Question 10

Solve the system of equations:

2x + y = 11\\ x - y = 1 \end{cases}$$ What is the solution $(x, y)$?
  1. (4,3)(4, 3)(4,3) (correct answer)
  2. (3,4)(3, 4)(3,4)
  3. (5,1)(5, 1)(5,1)
  4. (2,7)(2, 7)(2,7)

Explanation: Use elimination method by adding the equations. Adding 2x + y = 11 and x - y = 1 gives 3x = 12, so x = 4. Substitute x = 4 into x - y = 1: 4 - y = 1, so y = 3. The solution is (4, 3). Choice B reverses the x and y values.