How can I tell whether two algebraic expressions are equivalent?
Quick answer
Try rewriting both expressions using valid algebraic steps until they have the same form; that can establish equivalence. You can also substitute the same value for each variable in both expressions to check your work. If the results differ, the expressions are not equivalent. If they match—even at several values—that is only a check, not a general proof: different expressions can agree at selected inputs.
The question, the details, the next step.
I have two expressions and want to know if they mean the same thing. What should I try first?
First, simplify or rewrite the expressions, then compare the results. For example, compare x(x + 9) and x² + 9x. Distributing x across the parentheses gives x² + 9x, so the expressions match after a valid rewrite and are equivalent. Expressions can look different while representing the same quantity; factored and expanded forms are a common example. [S2]
Write down each step so you can check that you have applied the same valid algebraic rules throughout. Avoid deciding from appearance alone: one expression being longer, or having parentheses, does not by itself make it different in value.
What if I’m not sure how to simplify them? Can plugging in numbers help?
Yes—as a check. Substitute the same number for each variable in both expressions, then evaluate each one using the order of operations. For example, compare x(x + 9) and x² + 9x at x = 2:
- First expression:
2(2 + 9) = 22. - Second expression:
2² + 9(2) = 22.
The matching results are consistent with equivalence. Substitution means replacing each variable with a number and simplifying the result. [S1] But this one check does not establish that the expressions match for every possible value. Use it to catch mistakes, not as your only proof.
Can two expressions match for one value but still not be equivalent?
Yes. Consider x² + 1 and 2x. At x = 1, both give 2, but at x = 0, the first gives 1 and the second gives 0. The second check is a counterexample: because the results differ for that input, the expressions are not equivalent.
This illustrates the useful distinction: one mismatch is enough to disprove equivalence, while one match is only a successful check. Even several matching inputs are checks rather than a general proof. To establish equivalence, rewrite the expressions using valid algebraic steps and compare their resulting forms. [S1]
What’s a simple routine I can use on homework without guessing lots of numbers?
Use a two-part routine: simplify to seek a general comparison, then substitute a value or two to check your algebra. Here is a copyable record:
Expression A: x(x + 9)
Expression B: x² + 9x
Rewrite A: x² + 9x
Comparison: same form, so equivalent
Check at x = 2: 22 and 22
For a suspected non-match, look for a value that gives different results. If you find one, show both substitutions clearly; that gives a concrete reason the expressions are not equivalent. If your simplification gets stuck, record what you tried and ask a teacher or tutor to identify the next algebra rule. A handful of test values can help locate an error, but they cannot replace a valid general rewrite.
A practical checklist
Copy both expressions carefully, including signs, exponents, and parentheses.
Simplify or rewrite each expression with valid algebraic steps.
Compare the resulting forms rather than their original appearance.
Substitute the same value into both expressions to check arithmetic.
If the results differ for any input, show that counterexample; do not treat matching checks as proof.
A few more useful answers
How many values should I test?
There is no number of matching test values that by itself proves equivalence. Use substitutions as checks, and use valid algebraic rewriting for a general proof.
If one test gives different results, do I need to test more?
No. A single input where the expressions give different results is enough to show they are not equivalent. Write out both evaluations so the mismatch is clear.
What if the expressions look different after I simplify them?
Check each algebra step and see whether another valid rewrite can put them into comparable forms. If you cannot find one, show your work to a teacher or tutor rather than relying only on matching test values.
Sources
References used for this guide.
- [S1] OpenStax Prealgebra: Evaluate, Simplify, and Translate Expressions openstax.org · Checked
- [S2] OpenStax Algebra 1: Equivalent Expressions openstax.org · Checked