Grade 8 Math Practice Test: Systems of Linear Equations
- Systems of Linear Equations
- 12 questions
- 15 minutes
This test covers solving systems of two linear equations by graphing, substitution and elimination, deciding how many solutions a system has, and writing systems for word problems. Tip: check your answer by putting both values into both equations.
Tap an option to mark your answer. You can change your mind, or tap the same option again to leave the question blank. The correct answers and explanations appear when you finish the test.Tap an option to answer; tap it again to leave the question blank.
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Question 1
Which system of equations has the solution (2, 3)?
Correct answer: D
A solution must make both equations true. With x = 2 and y = 3: 2 + 3 = 5 and 2 − 3 = −1, so both equations hold. In the system with x − y = 1, the second equation fails, since 2 − 3 is −1, not 1.
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Question 2
The graph shows the lines y = x + 2 and y = −2x + 5.
What is the solution of the system?
Correct answer: D
The solution is the point where the lines cross, (1, 3). Check: 3 = 1 + 2 and 3 = −2(1) + 5. (0, 2) and (0, 5) are only the y-intercepts of the two lines.
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Question 3
Solve the system.
y = 2x
x + y = 12Correct answer: A
Substitute 2x for y: x + 2x = 12, so 3x = 12 and x = 4. Then y = 2 × 4 = 8. (8, 4) switches x and y, and (6, 6) does not satisfy y = 2x.
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Question 4
Solve the system.
3x + 2y = 16
3x − 2y = 8Correct answer: C
Add the equations to eliminate y: 6x = 24, so x = 4. Then 3(4) + 2y = 16 gives 2y = 4 and y = 2. Check in the second equation: 12 − 4 = 8.
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Question 5
Which system of equations has no solution?
Correct answer: B
Lines with the same slope and different y-intercepts are parallel and never meet, so the system has no solution. The system with 2y = 4x + 2 describes the same line twice and has infinitely many solutions.
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Question 6
How many solutions does this system have?
2x + y = 7
4x + 2y = 14Correct answer: B
The second equation is the first one multiplied by 2, so both describe the same line. Every point on that line is a solution, so there are infinitely many.
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Question 7
A school play sold 20 tickets for $133 in all. Adult tickets cost $8 and student tickets cost $5. How many adult tickets were sold?
Correct answer: C
Let a be adult tickets and s student tickets: a + s = 20 and 8a + 5s = 133. Substituting s = 20 − a gives 3a + 100 = 133, so a = 11. 9 is the number of student tickets.
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Question 8
The sum of two numbers is 30 and their difference is 6. Which system can be used to find the numbers x and y?
Correct answer: A
Sum means add and difference means subtract, so x + y = 30 and x − y = 6. The numbers are 18 and 12. xy = 6 would describe a product.
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Question 9
Gym A charges a $40 sign-up fee plus $20 per month. Gym B charges a $10 sign-up fee plus $25 per month. After how many months will the total cost be the same at both gyms?
Correct answer: B
Set the costs equal: 40 + 20m = 10 + 25m, so 30 = 5m and m = 6. Both gyms then cost $160.
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Question 10
Solve the system.
2x + 3y = 13
x − y = −1Correct answer: D
From the second equation, x = y − 1. Substitute: 2(y − 1) + 3y = 13, so 5y = 15 and y = 3, x = 2. (3, 2) works in the first equation but gives 3 − 2 = 1, not −1, in the second.
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Question 11
The two lines in a system of linear equations are parallel. What does this tell you about the system?
Correct answer: A
A solution is a point on both lines. Parallel lines never meet, so there is no such point and the system has no solution.
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Question 12
Solve the system and find the value of x + y.
x + 2y = 7
3x − y = 7Correct answer: C
From the first equation, x = 7 − 2y. Then 3(7 − 2y) − y = 7, so 21 − 7y = 7, y = 2 and x = 3. So x + y = 5.
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