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 \(5x-4y+8.\)