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: Add, subtract, multiply (non-zero), divide (non-zero) on both sides.

Common stuck point: Multiplying or dividing by zero is not a valid equivalence transformation — it destroys the equation or creates false solutions.

Sense of Study hint: After each step, ask: did I do exactly the same thing to both sides? If yes, the equation is still valid.

Worked Examples

Example 1

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

Solution

  1. 1
    Subtract 5 from both sides (an equivalence transformation): x = 7.
  2. 2
    The transformation preserves the solution set: the solutions of x + 5 = 12 and x = 7 are identical.
  3. 3
    Any value satisfying one equation satisfies the other.

Answer

x = 7
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?

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 = 15 to x = 5?

Example 2

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

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

equationsbalance principle