5-7 Equivalent Algebraic Expressions

Property

Two algebraic expressions are equivalent if they name the same number for all values of the variable.

Examples

• The expressions $4x + 2x$ and $6x$ are equivalent. If we test $x=3$, we get $4(3) + 2(3) = 12 + 6 = 18$, and $6(3) = 18$.
• The expressions $10-2x$ and $8x$ are not equivalent. If we test $x=1$, we get $10-2(1)=8$ and $8(1)=8$. But if we test $x=2$, we get $10-2(2)=6$ and $8(2)=16$. Since they are not equal for all values, they are not equivalent.
• The expressions $9y - 3y$ and $6y$ are equivalent. For any value of $y$, subtracting three $y$'s from nine $y$'s always results in six $y$'s.

Explanation

Think of equivalent expressions as two different ways to write the same value. No matter what number you substitute for the variable, they will always produce the same result because they are mathematically identical.

Property

Two algebraic expressions are equivalent if one can be transformed into the other using rules of arithmetic.

Commutative Property: The order does not change the result.
$A + B = B + A$ and $A \times B = B \times A$

Associative Property: The grouping does not change the result.
$A + (B + C) = (A + B) + C$ and $A \times (B \times C) = (A \times B) \times C$