Equivalent Expressions

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

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.

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. $4(10x-5)$

Answer

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?

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

Answer

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 $5x-4y+8.$