Practice Equivalence Transformation in Math

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

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Quick Recap

Operations applied to both sides of an equation that transform its form while leaving its solution set completely unchanged.

Whatever you do to one side, do to the other — the balance stays true.

Showing a random 20 of 50 problems.

Example 1

medium
Why must we NOT multiply both sides of an equation by 0?

Example 2

medium
Solve 3x=6\frac{3}{x}=6.

Example 3

medium
True or false: multiplying both sides of x=3x = 3 by (x−3)(x-3) yields an equivalent equation.

Example 4

hard
Explain why solving x2=4xx^2 = 4x by dividing both sides by xx loses a solution, and give the correct method.

Example 5

hard
Show step-by-step that 2x−13=x+24\frac{2x-1}{3} = \frac{x+2}{4} is equivalent to 5x=105x = 10.

Example 6

medium
Fill in: To remove the fraction in x3−1=4\frac{x}{3} - 1 = 4, you can ___ both sides by 3 first or after adding 1.

Example 7

challenge
For the equation x+5+x=5\sqrt{x+5} + \sqrt{x} = 5, explain why isolating one radical before squaring is necessary, and solve.

Example 8

easy
Solve x−8=11x - 8 = 11 using one equivalence transformation.

Example 9

hard
Solve 2x+1=x−1\sqrt{2x+1} = x - 1 and identify any extraneous solutions.

Example 10

easy
Solve x4=9\frac{x}{4} = 9.

Example 11

easy
Solve x+4=9x+4=9 using an equivalence transformation.

Example 12

medium
Why is multiplying both sides of 1x=2\frac{1}{x} = 2 by xx usually safe here?

Example 13

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Solve x3+2=5\frac{x}{3}+2=5.

Example 14

easy
Is adding 0 to both sides a valid equivalence transformation?

Example 15

easy
Solve 7x=497x = 49.

Example 16

hard
Solve the inequality −2x+5≥11-2x + 5 \geq 11.

Example 17

easy
Solve x+x=10x+x=10.

Example 18

easy
Which transformation converts x+7=12x + 7 = 12 to x=5x = 5?

Example 19

medium
Why is squaring both sides of an equation NOT always an equivalence transformation?

Example 20

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Solve −3x+7=1-3x + 7 = 1.
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