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Systems of equations — ACT practice questions

Practice 43 Systems of equations questions in the app

What this topic is

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.

Sample questions

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Question 1Easier

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

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Answer: C — 18

Let xx be the larger number and yy the smaller. Then x+y=28x+y=28 and x−y=8x-y=8. Adding the equations gives 2x=362x=36, so x=18x=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?

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Answer: C — 15

If ss and hh are the shirt and hat prices, 3s+2h=513s+2h=51 and s+4h=27s+4h=27. From the second, s=27−4hs=27-4h; substituting gives 81−10h=5181-10h=51, so h=3h=3 and s=15s=15.

Question 3Easier

Let ff be defined by f(x)=2x+1f(x)=2x+1, and let gg be defined by g(x)=x2−x−3g(x)=x^2-x-3. The graphs of y=f(x)y=f(x) and y=g(x)y=g(x) intersect at one of the following (x,y)(x,y) points. Which one?

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Answer: A — (4,9)(4,9)

Setting 2x+1=x2−x−32x+1=x^2-x-3 gives x2−3x−4=0x^2-3x-4=0, which factors as (x−4)(x+1)=0(x-4)(x+1)=0. At x=4x=4, y=2(4)+1=9y=2(4)+1=9, and checking x2−x−3=16−4−3=9x^2-x-3=16-4-3=9 confirms (4,9)(4,9) is an intersection point.

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Practice 43 Systems of equations questions in the app

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