Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for 7th grade equivalent expressions? Then this Grade 7 lesson on Rewriting Linear Expressions in Equivalent Forms explains each idea step by step, with worked examples.
Quick answer: Equivalent expressions have the same value for every value of the variable. You can create them with the distributive property, such as 3(x + 4) = 3x + 12, by combining like terms, and by factoring out a greatest common factor. Rewriting an expression can also reveal meaning, like p + 0.05p = 1.05p.
Standards: 7.EE.A.1, 7.EE.A.2 · Level: Grade 7
What makes two expressions equivalent?
Two expressions are equivalent when they give the same value for every number that replaces the variable. For example, 3(x + 4) and 3x + 12 both equal 18 when x = 2, and they match for every other value too.
Expressions that name the same quantity for all values of the variable. One test value can prove two expressions different, but it can never prove them equal.
How do you expand and combine like terms?
a(b + c) = ab + ac and a(b − c) = ab − ac
Like terms have the same variable part, so their coefficients can be added or subtracted. The terms 8x and 3x combine, but 8x and 3 do not.
| Expression | Expanded form |
|---|---|
| −2(3x − 5) | −6x + 10 |
| 12(8x + 6) | 4x + 3 |
| 0.4(5x − 2.5) | 2x − 1 |
Distribute to every term: 4(x + 3) is 4x + 12, not 4x + 3. A minus sign in front of parentheses flips every sign inside: −(x − 4) = −x + 4.
How do you factor a linear expression?
Factoring runs the distributive property backward. Find the greatest common factor (GCF) of the coefficients, write it outside the parentheses, and put what remains inside. For 12x + 18, the GCF is 6, so the result is 6(2x + 3). Expand to check.
How can rewriting an expression show its meaning?
A price p plus 5% tax is p + 0.05p. Combining like terms gives 1.05p, which says “multiply the price by 1.05” in one step.
- Distribute every factor, keeping each sign.
- Group like terms.
- Combine the coefficients.
- Check with a test value.
Simplify 4(2x − 3) − (x − 5).
8x − 12 − x + 5 = 7x − 7
Check with x = 2: 4(1) − (−3) = 7 and 7(2) − 7 = 7.
Circle every sign before you move a term. Most errors in this chapter are lost minus signs, not hard arithmetic.
- The distributive property multiplies a factor by every term inside the parentheses.
- Like terms share the same variable part, so only their coefficients are added or subtracted.
- Factoring out the greatest common factor reverses expanding, and expanding checks it.
- A percent change can be written as one multiplier: a 5% increase is 1.05 times the original.
Rewriting Linear Expressions in Equivalent Forms: frequently asked questions
How can I prove two expressions are not equivalent?
Find one value of the variable that gives different results. A single mismatch is enough, but matching values for a few numbers does not prove equivalence.
What happens when a negative number is outside the parentheses?
Multiply it by every term inside, changing signs as the sign rules require. For example, −3(x − 2) = −3x + 6 because −3 times −2 is positive.
Can I combine 5x and 5?
No. They are not like terms, because one has the variable x and the other does not. The sum 5x + 5 can only be factored as 5(x + 1).
What is the greatest common factor of 18x and 24?
It is 6, the largest number dividing both coefficients evenly. Then 18x + 24 = 6(3x + 4).
Why write p + 0.05p as 1.05p?
Because p is 1p, and 1p + 0.05p = 1.05p. One multiplication then gives the total price, with no separate tax step.
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