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ACT Math Help: Systems Of Equations

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

Question 1 / 10

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

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Question 1

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

2
4 (correct answer)
8
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 $12 for adults and $8 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?

$30$ (correct answer)
$45$
$60$
$90$

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

A taxi charges a $3 flat fee plus $2 per mile. Another taxi charges a $1 flat fee plus $3 per mile. For what number of miles $x$ do the two taxis cost the same (in dollars)?

\begin{cases} y = 2x + 3\\ y = 3x + 1 \end{cases}

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

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

If $2x + 3y = 13$ and $2x - 3y = 1$ , what is the value of $y$ ?

1
2 (correct answer)
3
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

Solve the system of equations:

\begin{cases} 5x - y = 14\\ 2x + y = 7 \end{cases}

What is the solution $(x, y)$ ?

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

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

Solve the system:

\begin{cases} x + 2y = 10\\ 3x - 2y = 6 \end{cases}

Which ordered pair satisfies both equations?

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

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

Consider the system of equations below: $3x + 2y = 12$ and $y = x - 4$ . What is the value of $x + y$ for the solution to this system?

0
2
4 (correct answer)
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

Consider the system of linear equations: $3x - y = 11$ and $2x + y = 9$ . What is the value of $x$ in the solution to this system?

$2$
$4$ (correct answer)
$5$
$8$

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

If the system below is true, what is the value of $y$ ?

\begin{cases} 3x + 4y = 1\\ 6x - 4y = 11 \end{cases}

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

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

What is the solution $(x, y)$ to the system $x + y = 6$ and $2x - y = 3$ ?

(4, 2)
(2, 4)
(1, 5)
(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).