Problem 81 - Introductory Algebra for College Students

Question

What are equivalent algebraic expressions?

Step-by-Step Solution

Answer

Equivalent algebraic expressions are expressions that simplify to the same value for any number substituted into the variables. They can be recognized by their identical variable terms and coefficients, and their equivalence can be verified by substitution.

Step 1: Understand the Concept

Equivalent algebraic expressions are expressions that simplify to the same number for any and all values substituted into any variables. For instance, the expressions \(2x + 3\) and \(3 + 2x\) are equivalent because no matter what number is substituted in for \(x\), both expressions will yield the same result.

Step 2: Recognize Equivalent Expressions

In practical terms, one can recognize equivalent expressions by noticing that they contain the same variables raised to the same powers, and identical numerical coefficients for these terms. It may not be immediately apparent if the terms are ordered differently, but rearranging the terms can clarify the equivalence. For example, given \(3x^2 - 2x + 1\) and \(1 - 2x + 3x^2\), rearranging the terms of the second expression gives \(3x^2 - 2x + 1\), proving the equivalence.

Step 3: Verify Equivalence

To confirm if two expressions are equivalent, one can substitute a number in for the variable and see if both expressions give the same result. For instance, with the expressions \(3x + 5\) and \(5 + 3x\), if a 2 is substituted in for \(x\), both expressions yield the result of 11, verifying that they are equivalent.