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Advanced Math / Equivalent expressions Difficulty: Medium

The expression $90 y Superscript 5 minus 54 y Superscript 4$ is equivalent to $r y Superscript 4 Baseline left parenthesis 15 y minus 9 right parenthesis$ , where $r$ is a constant. What is the value of $r$ ?

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Explanation

The correct answer is $6$ . Applying the distributive property to the expression $r y Superscript 4 Baseline left parenthesis 15 y minus 9 right parenthesis$ yields $15 r y Superscript 5 minus 9 r y Superscript 4$ . Since $90 y Superscript 5 minus 54 y Superscript 4$ is equivalent to $r y Superscript 4 Baseline left parenthesis 15 y minus 9 right parenthesis$ , it follows that $90 y Superscript 5 minus 54 y Superscript 4$ is also equivalent to $15 r y Superscript 5 minus 9 r y Superscript 4$ . Since these expressions are equivalent, it follows that corresponding coefficients are equivalent. Therefore, $90 equals 15 r$ and $- 54 = - 9 r$ . Solving either of these equations for $r$ will yield the value of $r$ . Dividing both sides of $90 equals 15 r$ by $15$ yields $6 equals r$ . Therefore, the value of $r$ is $6$ .