Equivalence Transformation Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Equivalence Transformation.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Equivalence transformation changes an equation's look while keeping the exact same solution set.

Common stuck point: The procedure for equivalence transformation is the easy part; the trap is changing only one side. Asking "Does this step act on both sides equally so the solution set is unchanged?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does this step act on both sides equally so the solution set is unchanged?

Worked Examples

Example 1

easy
Starting from x+5=12x + 5 = 12, apply an equivalence transformation to solve.

Answer

x=7x = 7

First step

1
Subtract 5 from both sides (an equivalence transformation): x=7x = 7.

Full solution

  1. 2
    The transformation preserves the solution set: the solutions of x+5=12x + 5 = 12 and x=7x = 7 are identical.
  2. 3
    Any value satisfying one equation satisfies the other.
An equivalence transformation changes the form of an equation without changing its solution set. Adding, subtracting, multiplying, or dividing both sides by a nonzero constant are all equivalence transformations.

Example 2

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

Example 3

medium
Show that 3(x−2)=93(x-2) = 9 and x−2=3x - 2 = 3 are equivalent equations.

Example 4

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

Example 5

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

Example 6

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 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.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
What equivalence transformation converts 3x=153x = 15 to x=5x = 5?

Example 2

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

Example 3

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

Example 4

easy
Solve x−3=7x-3=7.

Example 5

easy
Solve 3x=123x=12.

Example 6

easy
Solve x5=2\frac{x}{5}=2.

Example 7

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

Example 8

easy
Solve x+10=4x+10=4.

Example 9

easy
Solve 2x=−82x=-8.

Example 10

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

Example 11

medium
Solve 3x+5=203x+5=20.

Example 12

medium
Solve 5x−2=2x+75x-2=2x+7.

Example 13

medium
Solve x+12=4\frac{x+1}{2}=4.

Example 14

medium
Why can dividing both sides of x2=xx^2=x by xx lose a solution?

Example 15

medium
Solve 2(x−3)=102(x-3)=10.

Example 16

medium
Solve x=4\sqrt{x}=4 and check for extraneous roots.

Example 17

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

Example 18

challenge
Solve x+6=x\sqrt{x+6}=x and identify any extraneous solution.

Example 19

challenge
Show that multiplying both sides of an inequality −x<3-x<3 by −1-1 requires flipping the sign.

Example 20

challenge
Solve the system by an equivalence transformation: x+y=6x+y=6, x−y=2x-y=2.

Example 21

medium
Solve 4x+1=2x+94x+1=2x+9.

Example 22

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

Example 23

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

Example 24

easy
Solve 7x=497x = 49.

Example 25

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

Example 26

easy
Solve x+1.5=4x + 1.5 = 4.

Example 27

medium
Solve 6x−4=2x+126x - 4 = 2x + 12.

Example 28

medium
Solve 2x+35=3\frac{2x+3}{5} = 3.

Example 29

medium
Solve −3x+7=1-3x + 7 = 1.

Example 30

medium
Solve 0.2x+1=0.5x−20.2x + 1 = 0.5x - 2.

Example 31

medium
Solve x2+x3=5\frac{x}{2} + \frac{x}{3} = 5.

Example 32

medium
Solve 5(x+2)−3=2x+135(x+2) - 3 = 2x + 13.

Example 33

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

Example 34

hard
Solve x+2x−1=3\frac{x+2}{x-1} = 3.

Example 35

hard
Solve the system {2x+y=7x−y=−1\begin{cases}2x + y = 7 \\ x - y = -1\end{cases} using equivalence transformations.

Example 36

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

Example 37

hard
Solve ∣2x−3∣=7|2x - 3| = 7 using equivalence transformations.

Example 38

challenge
Find all xx satisfying x2−4x−2=5\frac{x^2 - 4}{x - 2} = 5 and state any restrictions.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

equationsbalance principle