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Question 11 • Algebra I

A math class is having a discussion on how to determine if the expressions 4⁢x−x+5 and 8−3⁢x−3 are equivalent. The class has suggested four different methods. Which describes the correct method?

Answer

Both expressions should be evaluated by substituting any two values for x . If for each substituted value, the final values of the expressions are the same, then the two expressions must be equivalent.

Both expressions should be evaluated by substituting any two values for x . If for each substituted value, the final values of the expressions are positive, then the two expressions must be equivalent.

Both expressions should be evaluated by substituting with one value for x . If the final values of the expressions are the same, then the two expressions must be equivalent.

Both expressions should be evaluated by substituting one value for x . If the final values of the expressions are both positive after the substitution, then the two expressions must be equivalent.

Explanation

Simplifying gives 4x − x + 5 = 3x + 5 and 8 − 3x − 3 = 5 − 3x. These are linear expressions, so if they give the same result for two different values of x, their difference—a linear expression—has two roots and must be zero for every x. Here, for example, at x = 1 the results are 8 and 2, so the expressions are not equivalent. Checking whether the results are positive, as in the other methods, does not establish equivalence.

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