Proving expressions are equivalent
Use structure to justify an identity and a counterexample to reject one.
Before you begin
- Distributing and collecting terms
- Substituting values
Require agreement for every allowed value
Equivalent expressions name the same value for every value of their variables in the stated domain. For example, 2(x + 3) and 2x + 6 are equivalent because distribution transforms one into the other. A pair that matches only for a special input is not an identity. Simplifying each expression is a stronger method than trying a handful of random inputs.
Use a counterexample efficiently
One input producing different outputs proves that expressions are not equivalent. For example, x² and 2x both equal 4 at x = 2, but at x = 3 they give 9 and 6. To show equivalence, explain a valid sequence of operations; to disprove it, show and calculate one counterexample. Distinguish these two types of mathematical argument.
Worked example
Are 3(x + 2) + x and 4x + 6 equivalent? Justify your answer.
Show the worked solution
- Distribute three: 3(x + 2) + x = 3x + 6 + x.
- Combine the like terms 3x and x to obtain 4x + 6.
- The steps use properties valid for every x, so the expressions are equivalent.
Yes; both simplify to 4x + 6.
Try it yourself
Which observation proves x + 4 and 4x are not equivalent?
Common mistakes
- Matching at one input does not establish an identity.
- One counterexample is sufficient to disprove equivalence.
What can you explain now?
Is 5(n + 1) equivalent to 5n + 1? Give a structural reason and a counterexample.
Compare with the explanation
No. 5(n + 1) = 5n + 5; at n = 0 the expressions give 5 and 1.
The outside factor multiplies both bracket terms. The unequal outputs at zero independently disprove the proposed identity.
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Bring the ideas together
Expressions and structure
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Common Core mathematics standards ↗