Practice Systems of Equations in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

Two or more equations sharing the same variables, where the solution must satisfy all equations simultaneously.

Where two lines cross—the point that satisfies both equations.

Showing a random 20 of 50 problems.

Example 1

challenge
For what kk does the system 2x+ky=42x + ky = 4, kx+8y=8kx + 8y = 8 have infinitely many solutions?

Example 2

medium
Solve: 5x+3y=75x + 3y = 7, 2x−3y=142x - 3y = 14.

Example 3

hard
Solve: x3+y2=4\frac{x}{3} + \frac{y}{2} = 4, x2−y3=1\frac{x}{2} - \frac{y}{3} = 1.

Example 4

hard
Solve: 2x+3y=122x + 3y = 12 and 4x−y=54x - y = 5.

Example 5

hard
Mixing problem: a chemist combines a 20%20\% acid and a 50%50\% acid solution to make 3030 L of 40%40\% acid. How many liters of each?

Example 6

medium
Solve by graphing (where do they meet?): y=x+1y = x + 1, y=−x+5y = -x + 5.

Example 7

easy
Solve: x=4x = 4, y=x−1y = x - 1.

Example 8

medium
At a fair, 33 child tickets and 22 adult tickets cost $31. 55 child and 44 adult tickets cost $57. Find each price.

Example 9

easy
Solve by substitution: y=2xy = 2x, x+y=6x + y = 6.

Example 10

easy
Solve by substitution: y=3xy = 3x, 2x+y=102x + y = 10.

Example 11

easy
Solve: 3x+y=113x + y = 11, y=x−1y = x - 1.

Example 12

easy
Solve: y=4y = 4, 2x+y=142x + y = 14.

Example 13

medium
How many solutions does the system x−2y=4x - 2y = 4, −2x+4y=−8-2x + 4y = -8 have?

Example 14

hard
Find (x,y)(x, y): 3x+4y=223x + 4y = 22, 5x−2y=−75x - 2y = -7.

Example 15

medium
Solve: x2+y=5\frac{x}{2} + y = 5, x−y=1x - y = 1.

Example 16

easy
Is (1,2)(1, 2) a solution of x+y=3x + y = 3 and x−y=−1x - y = -1?

Example 17

easy
Solve: x+y=7x + y = 7, x−y=3x - y = 3.

Example 18

easy
Two lines have the same slope but different intercepts. How many solutions?

Example 19

easy
Solve: x+2y=8x + 2y = 8, x−y=2x - y = 2.

Example 20

easy
Solve the system: x+y=10x + y = 10 and x−y=4x - y = 4.
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