Equivalent algebraic expressions have the same value for every permissible value of their variables, even when their terms or arrangement look different. Equivalence is recognized through the commutative, associative, and distributive properties, including rewriting as and combining like terms, rather than by checking only one substitution. The focus is on simplifying and comparing linear and basic polynomial expressions, not advanced polynomial identities or more abstract algebraic generalizations.
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:
Example: Are and equivalent?
Simplify the first expression:
Use the distributive property to multiply by each term inside the parentheses:
Combine the like terms and :
The simplified form is exactly the second expression:
Therefore, the expressions are equivalent. This works for every value of , not just one value you might substitute.
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