Systems of equations — ACT practice questions

49 practice questions in the app
Free

Practice 49 Systems of equations questions in the app

What this topic is

Systems of equations ask for values that satisfy two equations at the same time.

Substitution works well when one equation already isolates a variable, while elimination, adding or subtracting the equations to cancel a variable, suits pairs where the coefficients line up easily. Graphically, the solution is the point where the two graphs cross, and a line and a parabola can cross at two points, one point or none.

ACT Mathematics often frames systems as word problems about two items bought at different prices or two numbers with a given sum and difference. Plugging the answer choices back into both equations is a dependable check.

Sample questions

Pick an answer to see whether it is right — nothing is saved, and nothing needs an account.

Question 1Easier

The sum of 2 numbers is 28. The larger number is 8 more than the smaller number. What is the larger number?

8✗ Not this one10✗ Not this one18✓ Correct28✗ Not this one

Show answer

Answer: C — 18

Let $x$ be the larger number and $y$ the smaller. Then $x + y = 28$ and $x - y = 8$ . Adding the equations gives $2 x = 36$ , so $x = 18$ .

Question 2Easier

Lee bought 3 shirts and 2 hats for $51. At the same prices, Kai bought 1 shirt and 4 hats for $27. What is the cost, in dollars, of 1 shirt?

9✗ Not this one12✗ Not this one15✓ Correct18✗ Not this one

Show answer

Answer: C — 15

If $s$ and $h$ are the shirt and hat prices, $3 s + 2 h = 51$ and $s + 4 h = 27$ . From the second, $s = 27 - 4 h$ ; substituting gives $81 - 10 h = 51$ , so $h = 3$ and $s = 15$ .

Question 3Easier

Let $f$ be defined by $f (x) = 2 x + 1$ , and let $g$ be defined by $g (x) = x^{2} - x - 3$ . The graphs of $y = f (x)$ and $y = g (x)$ intersect at one of the following $(x, y)$ points. Which one?

(4, 9)

✓ Correct

(- 1, 3)

✗ Not this one

(0, - 3)

✗ Not this one

(3, 7)

✗ Not this one

Show answer

Answer: A — $(4, 9)$

Setting $2 x + 1 = x^{2} - x - 3$ gives $x^{2} - 3 x - 4 = 0$ , which factors as $(x - 4) (x + 1) = 0$ . At $x = 4$ , $y = 2 (4) + 1 = 9$ , and checking $x^{2} - x - 3 = 16 - 4 - 3 = 9$ confirms $(4, 9)$ is an intersection point.

Every question is picked for where you are right now, wrong answers come back until they stick, and it all works offline.

Practice 49 Systems of equations questions in the app

Systems of equations — ACT practice questions

What this topic is

Sample questions

More ACT topics

Mathematics