Systems of equations — ACT practice questions

Systems of equations on the ACT Mathematics test covers two relations that share a common solution, including linear pairs or a line in function notation meeting a quadratic. A student has to write equations from a word problem, or treat a graph intersection as that shared solution, then solve by substitution or elimination. Questions usually ask for one unknown, such as a price or the larger of two numbers, or for the ordered pair where the graphs meet. Answer choices are typically numbers or coordinate pairs. Common traps include solving for the wrong variable, swapping the two quantities, and treating an x-value as the intersection point.

Section: Mathematics 45 questions on the paper
43 practice questions in the app

Sample questions

Question 1

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

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 $2x=36$ , so $x=18$ .

Question 2

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?

Show answer

Answer: C — 15

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

Question 3

Let $f$ be defined by $f(x)=2x+1$ , 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?

A $(4,9)$
B $(-1,3)$
C $(0,-3)$
D $(3,7)$

Show answer

Answer: A — $(4,9)$

Setting $2x+1=x^2-x-3$ gives $x^2-3x-4=0$ , 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.

Practice 43 Systems of equations questions in the app

Systems of equations — ACT practice questions

Sample questions

More ACT topics

Mathematics