Skill: Recognize equivalent algebraic expressions

Two algebraic expressions are equivalent if they have the same value for every allowed value of the variable. To recognize this, simplify both expressions using:

• The distributive property: $a(b+c)=ab+ac$
• Combining like terms: terms with the same variable part
• The commutative and associative properties to rearrange or regroup terms

Example: Are $3(x+4)+2x$ and $5x+12$ equivalent?

Simplify the first expression:

$$3(x+4)+2x$$

Use the distributive property to multiply $3$ by each term inside the parentheses:

$$3x+12+2x$$

Combine the like terms $3x$ and $2x$:

$$5x+12$$

The simplified form is exactly the second expression:

$$3(x+4)+2x=5x+12$$

Therefore, the expressions are equivalent. This works for every value of $x$, not just one value you might substitute.