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Equivalent expressions worksheets

Two expressions are equivalent when they name the same number whatever you put in for x. These free printable worksheets give one expression and four candidates, or a pair to judge — and every wrong option is a mistake a real student makes.

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Equivalent expressions
Equivalent expressions worksheet — Brackets, like terms, factoring, free printable with answer key

The answer key prints on a separate page.

What's on this sheet

Multiplying a bracket out and collecting like terms are the sheets before this one; this is the step where a student checks their own work. Every distractor here has a name: 3(x + 2) written as 3x + 2 is the bracket multiplied once instead of twice, 2x + 3 written as 5x is a number added to a term as though it were one, and 4(x − 1) written as 4x + 4 is the minus lost on the way through.

Why this sheet works

A page of plausible wrong answers is worth more than a page of obviously wrong ones, because the second kind is answered by elimination and teaches nothing. Every option is built from the same numbers as the question for that reason — an option holding a number that appears nowhere can be crossed off without doing any algebra at all.

How this one works

Which is the same as 6x + 2 + x?

6x + 2 + x≡?
6x + 2 + x=7x + 2
6x + 2 + x=7x + 2
not:9x
x = 3
6x + 2 + x→23
7x + 2→23
x = 1:6x → 66x² → 6
x = 0:6x + 2 + x → 2every x gone
x = 3:6x + 2 + x → 237x + 2 → 23
  1. Equivalent does not mean it looks similar — it means the two expressions give the SAME answer whichever number x turns out to be. That is a strong claim, and it is checkable.
  2. Only the x terms can join: 6x and the bare x are 7 lots of x. The 2 is not an x, so it cannot go in with them and stays where it is.
  3. Sweeping the 2 in as well would give 9x, and that is the commonest wrong answer here — 2 is 2, whatever x turns out to be.
  4. Check it by putting a number in — the same number in both. With x = 3: 6x + 2 + x is 23, and 7x + 2 is 23. Equal, so the two agree at least here.
  5. That is how to test any of the choices: put the same x into each, and only the equivalent one comes out equal. Pick something other than 0 and 1, though — at 1 every power of x is 1, so 6x and 6x² both come out 6; and at 0 every x term vanishes, so two expressions that differ only in their x part look identical. That is why the check above used 3.

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