Grade 8 Math Practice Test: Multi-Step Equations with Fractions, Decimals and Variables on Both Sides
- Multi-Step Equations and Number of Solutions
- 10 questions
- 15 minutes
This test covers linear equations that need the distributive property, combining like terms and moving variables from both sides, including fractions and decimals. Tip: if the variable terms cancel, a true statement like 6 = 6 means infinitely many solutions and a false one like 0 = 17 means no solution.
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
Solve for x.
12(8x − 6) = 3(x + 5)
Correct answer: C
Distribute: 4x − 3 = 3x + 15. Subtract 3x and add 3: x = 18. x = 8 forgets to multiply the 5 by 3.
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Question 2
Solve for x.
0.4(x + 5) = 0.2x + 3
Correct answer: A
Distribute: 0.4x + 2 = 0.2x + 3. Then 0.2x = 1, so x = 1 ÷ 0.2 = 5. x = −10 comes from writing 0.4x + 5, multiplying only the x by 0.4.
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Question 3
Solve for x.
x3 + x4 = 14
Correct answer: D
Multiply every term by 12: 4x + 3x = 168, so 7x = 168 and x = 24. Check: 8 + 6 = 14. x = 2 divides 14 by 7 without multiplying by 12.
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Question 4
How many solutions does this equation have?
3(2x − 4) = 6x + 5
Correct answer: B
The left side is 6x − 12. Subtracting 6x from both sides leaves −12 = 5, which is never true, so there is no solution. Same x-terms with different constants always give no solution.
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Question 5
How many solutions does this equation have?
5x − 2(x − 3) = 3x + 6
Correct answer: D
The left side is 5x − 2x + 6 = 3x + 6, exactly the right side. The equation is true for every x, so there are infinitely many solutions. Forgetting that −2 · −3 = +6 would wrongly suggest no solution.
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Question 6
For which value of c does the equation 4(x + 3) = 4x + c have infinitely many solutions?
Correct answer: D
The left side is 4x + 12. The two sides are identical only when c = 12. Any other value gives no solution, and c = 3 forgets to multiply the 3 by 4.
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Question 7
For which value of a does the equation ax + 7 = 5x − 2 have no solution?
Correct answer: C
With a = 5 the x-terms cancel and 7 = −2 is false, so there is no solution. For any other a, including −5, the x-terms do not cancel and there is exactly one solution.
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Question 8
Candle A is 20 cm tall and burns 1.5 cm per hour. Candle B is 14 cm tall and burns 0.75 cm per hour. Both are lit at the same time. After how many hours are they the same height?
Correct answer: B
Set the heights equal: 20 − 1.5t = 14 − 0.75t. Then 6 = 0.75t, so t = 8 hours; both candles are 8 cm tall then. 4 hours divides 6 by 1.5 and ignores that candle B also burns.
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Question 9
Solve for x.
2(3x − 1) − 4x = 5x − 11
Correct answer: A
Left side: 6x − 2 − 4x = 2x − 2. So 2x − 2 = 5x − 11, which gives 9 = 3x and x = 3. x = 10/3 comes from writing 6x − 1, forgetting to multiply the 1 by 2.
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Question 10
Solve for x.
2x + 13 = x − 42
Correct answer: A
Multiply both sides by 6: 2(2x + 1) = 3(x − 4), so 4x + 2 = 3x − 12 and x = −14. x = −5 multiplies only the first terms, giving 4x + 1 = 3x − 4.
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