ACT Math Quiz: Systems Of Equations

What this quiz covers

This quiz focuses on Systems Of Equations, giving you a quick way to practice the rules, question types, and explanations that matter most for ACT Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

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

1. $30$ (correct answer)
2. $45$
3. $60$
4. $90$

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

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

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

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

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

Question 5

1. $(3, 1)$ (correct answer)
2. $(2, 3)$
3. $(1, 3)$
4. $(3, -1)$

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 6

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

Explanation: Use elimination: add equations $x + 2y = 10$ and $3x - 2y = 6$. The y terms cancel, giving $4x = 16$, so $x = 4$. Substitute $x = 4$ into $x + 2y = 10$: $4 + 2y = 10$, so $y = 3$. The solution is $(4, 3)$.

Question 7

1. 0
2. 2
3. 4 (correct answer)
4. 8

Explanation: This is a systems of equations question testing substitution. Choice C (4) is correct — substitute y = x − 4 into the first equation: 3x + 2(x − 4) = 12 → 5x − 8 = 12 → 5x = 20 → x = 4. Plug back in: y = 4 − 4 = 0. Therefore x + y = 4 + 0 = 4. Choice A (0) reports only y = 0 rather than x + y. Choice B (2) results from an arithmetic error in solving 5x = 20 — possibly dividing by 10 instead of 5. Choice D (8) finds x = 4 correctly but then substitutes incorrectly into the second equation, using y = x instead of y = x − 4, giving y = 4 and x + y = 8. Pro tip: After finding one variable in a system, always substitute back into the equation that contains BOTH variables to find the second. And re-read what the question asks — here it's asking for x + y, not just x or y individually.

Question 8

1. $2$
2. $4$ (correct answer)
3. $5$
4. $8$

Explanation: The correct answer is B (4). Add the two equations to eliminate y: (3x − y) + (2x + y) = 11 + 9 → 5x = 20 → x = 4. Check: substitute x = 4 into the second equation: 2(4) + y = 9 → y = 1. Verify in first: 3(4) − 1 = 11 ✓. A (2) confuses x and y — the student may find y = 1 and misread it. C (5) results from an arithmetic error when adding the equations: computing 11 + 9 = 25 instead of 20. D (8) comes from a substitution error. Elimination by addition works here because the y terms (+y and −y) cancel perfectly.

Question 9

1. $-\tfrac{5}{4}$
2. $-\tfrac{3}{4}$ (correct answer)
3. $\tfrac{5}{4}$
4. $\tfrac{3}{4}$

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

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

Explanation: Use elimination to solve this system. Adding the equations x + y = 6 and 2x - y = 3 eliminates y: 3x = 9, so x = 3. Substituting x = 3 into the first equation: 3 + y = 6, so y = 3. The solution is (3, 3).

Question 11

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

Explanation: Use elimination to solve this system. Adding $x + y = 8$ and $x - y = 4$ eliminates y: $2x = 12$, so $x = 6$. Substituting into $x + y = 8$: $6 + y = 8$, so $y = 2$. Therefore, $x = 6$.

Question 12

1. 2
2. 3
3. 9 (correct answer)
4. There is no such value of $k$.

Explanation: Systems of equations have no solution when the lines are parallel—same slope but different y-intercepts. Convert the first equation to slope-intercept form: $y = -2x/3 + 2$ (slope = $-2/3$). For the second equation: $y = -6x/k + 7/k$ (slope = $-6/k$). Set the slopes equal: $-6/k = -2/3$, which gives $k = 9$. Verify the y-intercepts are different: $2 ≠ 7/9$ ✓, confirming the lines are parallel. This is a conceptual question testing whether you understand what "no solution" means geometrically.

Question 13

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

Explanation: Use elimination: Add the equations $x + y = 8$ and $3x - y = 4$. This gives $4x = 12$, so $x = 3$. Substitute $x = 3$ into $x + y = 8$: $3 + y = 8$, so $y = 5$. The solution is $(3, 5)$.

Question 14

1. $(1, 5)$
2. $(2, 1)$ (correct answer)
3. $(3, -3)$
4. $(1, -5)$

Explanation: Use elimination: Add the equations $4x + y = 9$ and $2x - y = 3$. This gives $6x = 12$, so $x = 2$. Substitute $x = 2$ into $2x - y = 3$: $4 - y = 3$, so $y = 1$. The solution is $(2, 1)$.

Question 15

1. $(2,3)$
2. $(6,2)$ (correct answer)
3. $(8,1)$
4. $(4,\tfrac{7}{2})$

Explanation: Set up the system where $x$ is flat fee and $y$ is per-mile rate: $x + 4y = 14$ and $x + 7y = 20$. Subtract the first from the second: $3y = 6$, so $y = 2$. Substitute back: $x + 4(2) = 14$, so $x = 6$. The solution is $(6, 2)$.

Question 16

1. 2
2. 3
3. 4 (correct answer)
4. 5

Explanation: Use the substitution method from 2x - y = 1, so y = 2x - 1, into 4x + 2y = 18. This gives 4x + 2(2x - 1) = 18, 4x + 4x - 2 = 18, 8x = 20, x = 2.5, y = 2(2.5) - 1 = 5 - 1 = 4. The value of y is 4. Choice B of 3 might come from solving 8x = 18 incorrectly.

Question 17

1. 0
2. 2
3. 4 (correct answer)
4. 8

Explanation: This is a systems of equations question testing substitution. Choice C (4) is correct — substitute y = x − 4 into the first equation: 3x + 2(x − 4) = 12 → 5x − 8 = 12 → 5x = 20 → x = 4. Plug back in: y = 4 − 4 = 0. Therefore x + y = 4 + 0 = 4. Choice A (0) reports only y = 0 rather than x + y. Choice B (2) results from an arithmetic error in solving 5x = 20 — possibly dividing by 10 instead of 5. Choice D (8) finds x = 4 correctly but then substitutes incorrectly into the second equation, using y = x instead of y = x − 4, giving y = 4 and x + y = 8. Pro tip: After finding one variable in a system, always substitute back into the equation that contains BOTH variables to find the second. And re-read what the question asks — here it's asking for x + y, not just x or y individually.

Question 18

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 19

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

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

Question 20

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).