"Equivalent expressions" is searched about 4,600 times a month in the United States, and the question behind it is nearly always a multiple-choice item: which expression is equivalent to 4(2x − 3) + 5x? The idea is the backbone of Grade 6 and 7 algebra (6.EE.3 and 6.EE.4, 7.EE.1), and the same skill is called simplifying and expanding in the UK, Australia and India. This page gives the definition, the two ways to test a pair, the five properties that produce equivalent expressions, the traps, ten worked examples and a printable sheet of 20 practice questions with answers.
Download the equivalent expressions sheet with 20 questions (PDF) (A4, the properties, the test and the worked examples on page 1, 20 practice questions with answers on page 2, free).
What "equivalent" means
Two expressions are equivalent when they give the same value for every value of the variable. Not for one value, not for most: for all of them.
- 2(x + 3) and 2x + 6: x = 5 gives 16 and 16; x = 0 gives 6 and 6; x = −4 gives −2 and −2. Equivalent, and the distributive property says why.
- x² and 2x: x = 2 gives 4 and 4, so they match once. But x = 3 gives 9 and 6. Not equivalent. One disagreement is enough.
An equivalent expression is the same quantity written differently, the way ½ and 0.5 and 50% are the same number. Simplifying, expanding, factoring and collecting like terms all produce equivalent expressions; that is what they are for.
Two ways to tell
1. Substitute (the quick test). Pick a value, preferably not 0 or 1 (they hide too many differences), put it into both expressions and compare.
- If the values differ: not equivalent. Proven.
- If the values match: probably equivalent, but not yet proven (x² and 2x matched at x = 2). Try a second value, or move to method 2.
On a multiple-choice question, substituting x = 2 into the question and every option eliminates the wrong ones in a minute.
2. Simplify both (the proof). Expand brackets, combine like terms, write the terms in the same order. If both reach the same expression, they are equivalent for every value, because each step used a property that holds for all numbers.
4(2x − 3) + 5x → 8x − 12 + 5x → 13x − 12. Any option that simplifies to 13x − 12 is equivalent; the others are not.
The properties that make equivalent expressions
| Property | Rule | Example |
|---|---|---|
| Commutative (addition, multiplication) | a + b = b + a; ab = ba | 3 + x = x + 3; 5y = y·5 |
| Associative | (a + b) + c = a + (b + c); (ab)c = a(bc) | (x + 2) + 7 = x + 9 |
| Distributive | a(b + c) = ab + ac; a(b − c) = ab − ac | 3(x + 4) = 3x + 12; 2(5y − 1) = 10y − 2 |
| Combining like terms (distributive in reverse) | ax + bx = (a + b)x | 3x + 5x = 8x; 7y − 2y = 5y |
| Factoring (distributive in reverse) | ab + ac = a(b + c) | 6x + 9 = 3(2x + 3) |
| Identity | a + 0 = a; 1·a = a | x + 0 = x |
| Inverse | a + (−a) = 0; a·(1/a) = 1 | 4x − 4x = 0 |
| Multiplying by a negative | −(a + b) = −a − b | −(x − 5) = −x + 5 |
Like terms have the same variable to the same power: 3x and −7x are like; 3x and 3x² are not; 4xy and −xy are like; constants are like each other. Only like terms combine.
Not equivalent, and not a property: subtraction and division are not commutative (5 − x ≠ x − 5; 10 ÷ x ≠ x ÷ 10).
Ten worked examples
- 3(x + 4) = 3x + 12. Distributive.
- 2x + 3x + x = 6x. Like terms (the lone x has coefficient 1).
- 5(2y − 3) + 4 = 10y − 15 + 4 = 10y − 11.
- 4(2x − 3) + 5x = 8x − 12 + 5x = 13x − 12.
- −2(x − 6) = −2x + 12. The negative multiplies both terms and flips the sign of the second.
- 7a − (3a + 2) = 7a − 3a − 2 = 4a − 2. The minus applies to everything in the brackets.
- 6x + 9: factor out 3: 3(2x + 3). Expand to check: 6x + 9.
- x·x·x = x³, not 3x. 3x is x + x + x.
- ½(8m + 4) = 4m + 2.
- 2(x + 3) + 3(x − 1) = 2x + 6 + 3x − 3 = 5x + 3.
The six traps
| Wrong | Right | Why |
|---|---|---|
| 3(x + 2) = 3x + 2 | 3x + 6 | distribute to every term |
| (x + 3)² = x² + 9 | x² + 6x + 9 | squaring a sum has a middle term |
| x + x = x² | 2x | adding gives a coefficient; multiplying gives a power |
| 2x · 3x = 6x | 6x² | multiply coefficients and multiply variables |
| 5 − (x + 2) = 5 − x + 2 | 3 − x | the minus applies to both terms |
| 3x + 2y = 5xy | cannot be simplified | unlike terms do not combine |
Each trap fails the substitution test at once: (1 + 3)² = 16 but 1² + 9 = 10.
Where it sits in each curriculum
| Curriculum | When | What is expected |
|---|---|---|
| US Common Core | Grade 6 (6.EE.3: apply the properties of operations to generate equivalent expressions; 6.EE.4: identify when two expressions are equivalent), Grade 7 (7.EE.1: add, subtract, factor and expand linear expressions with rational coefficients), Grade 8 and Algebra 1 (with exponents and more variables) | distributive property, like terms, factoring a common factor |
| England | KS3 (Year 7 and 8: simplify, expand single brackets, factorise), GCSE (expand double brackets, factorise quadratics, identities with the ≡ sign) | the word "identity" and the ≡ symbol at GCSE |
| Australia | Year 7 (extend and apply the laws and properties of arithmetic to algebraic terms), Year 8 (expand and factorise) | simplify, expand, factorise |
| India (CBSE) | Class 7 (Algebraic Expressions: like terms, adding and subtracting), Class 8 (Algebraic Expressions and Identities: multiplication, the standard identities) | expansion and the identities |
20 practice questions (answers in the PDF)
For each, simplify or decide whether the pair is equivalent.
- 5(x + 3) 2. 4y + 7y − 2y 3. 2(3a − 4) + 5 4. Is 6(x − 2) equivalent to 6x − 2? 5. −3(2m + 5) 6. 8x − (2x − 7) 7. Factor 10y + 15 8. Is x + x + x equivalent to x³? 9. ¼(12p − 8) 10. 3(x + 1) + 2(x + 4) 11. Is (x + 4)² equivalent to x² + 16? 12. 5k · 2k 13. 9 − (n + 4) 14. Is 2(x + y) equivalent to 2x + y? 15. 7w − 3 + 2w + 8 16. Factor 4x² + 6x 17. Is 3 + 5x equivalent to 5x + 3? 18. 6(2t − 1) − 4t 19. Is 10 ÷ x equivalent to x ÷ 10? 20. Which is equivalent to 4(2x − 3) + 5x: (a) 13x − 3 (b) 13x − 12 (c) 8x − 7 (d) 13x + 12?
Download the sheet and the 20 questions with answers (PDF)
Where to get help
A 1-on-1 online maths class with a tutor who holds a master's in mathematics and teaches Grade 6 to 8 and Algebra 1, KS3 and GCSE, the Australian Years 7 and 8 and CBSE Class 7 and 8 every week: algebra taught with the substitution check as a habit, so the child catches a wrong expansion before the teacher does. The Grade 6 maths page lists the units; the free 30-minute first class is a real lesson on this week's homework. Related pages: the algebra formulas list, how to explain equations to a child and why am I bad at algebra.
Questions parents ask
1What are equivalent expressions?
2How do you know if two expressions are equivalent?
3Is 3(x + 2) equivalent to 3x + 2?
4Is (x + 3)² equivalent to x² + 9?
5What properties make expressions equivalent?
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