Question
Fill in the blank to make equivalent rational expressions. \( \frac{5}{y+7}=\frac{\square}{(y+9)(y+7)} \)
Ask by Logan Willis. in the United States
Jan 11,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
To make the rational expressions equivalent, replace the blank with \(5(y + 9)\).
Solution
To make the rational expressions equivalent, you need to ensure that both fractions have the same value for all permissible values of \( y \).
Given:
\[
\frac{5}{y+7} = \frac{\square}{(y+9)(y+7)}
\]
To find the value that should replace the blank, observe that the denominator on the right side \((y+9)(y+7)\) is just \((y+7)\) multiplied by an additional factor \((y+9)\). To maintain equality, the numerator on the right side must also be multiplied by the same additional factor.
Therefore, multiply the numerator \(5\) by \((y+9)\):
\[
5 \times (y + 9) = 5(y + 9)
\]
So, the equivalent rational expressions are:
\[
\frac{5}{y+7} = \frac{5(y+9)}{(y+9)(y+7)}
\]
**Answer:** \(5(y + 9)\)
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Simplify this solution The Deep Dive
To make the rational expressions equivalent, we need to scale the numerator of the left-hand side expression so that the denominators match. Since the denominator on the right side is \((y + 9)(y + 7)\), we can multiply the left-side numerator \(5\) by \((y + 9)\). Therefore, the blank should be filled with: \(5(y + 9)\) Now the equivalent expression is: \(\frac{5}{y + 7} = \frac{5(y + 9)}{(y + 9)(y + 7)}\).
