# Equivalent Expressions

**Equivalent expressions** are mathematical statements that look different but have the same value.

## Introduction

Algebraic expressions, such as \(4x+3y-2w^2,\) contain variables, numbers, and mathematical operations. Algebraic expressions may be written in different ways, but still mean the same thing.

For example, the expressions \[r+r+r+r \text{ and } 4r\]

are equivalent. Regardless of what number is substituted for \(r,\) the two expressions will have the same value.

## Identifying Equivalent Expressions

We can use properties of arithmetic and the combining of like terms to write equivalent expressions.

## Which expressions are equivalent to \(5(8x-4)\,?\)

A. \(40x - 20\)

B. \(20x - 20 + 20x\)

C. \((8x-4)5\)

D. \(4(10x-5)\)

All of the expressions are equivalent to \(5(8x-4).\)When we distribute the 5, we get \(5(8x) - 5(4) = 40x - 20,\) or choice A.

For choice B, we can combine the like terms of \(20x\) and \(20x\) to get \(40x - 20.\)

In choice C, we can distribute the 5 to get \(40x - 20.\)

In choice D, when we distribute the 4, we get \(4(10x) - 4(5) = 40x - 20.\)

## Joe wrote two expressions that he said were equivalent and connected them with an equal sign, as shown below. Are his two expressions equivalent?

\[\frac{15x-12y+24}{3} = 5x - 12y + 8\]

Joe's two expressions of \(\frac{15x-12y+24}{3}\) and \(5x - 12y + 8\) are not equivalent. In order to simplify the expression \(\frac{15x-12y+24}{3},\) every term in the numerator of the fraction needs to divided by 3, yielding an equivalent expression of \(5x-4y+8.\)

**Cite as:**Equivalent Expressions.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/equivalent-expressions/