Question
Write the rational expression with the indicated denominator. 19/(z^2-7z)=?/z(z-7)(z-5). Solve for the question mark
Answers
GPT-4o mini
To find the value of the question mark in the rational expression \( \frac{19}{z^2 - 7z} = \frac{?}{z(z - 7)(z - 5)} \), we first need to simplify \( z^2 - 7z \).
We can factor \( z^2 - 7z \) as follows:
\[
z^2 - 7z = z(z - 7)
\]
Now we set up the equation:
\[
\frac{19}{z(z - 7)} = \frac{?}{z(z - 7)(z - 5)}
\]
To find the numerator represented by the question mark, we can cross multiply:
\[
19 \cdot (z(z - 7)(z - 5)) = ? \cdot (z(z - 7))
\]
This means we need to express \( 19(z(z - 7)(z - 5)) \) divided by \( z(z - 7) \):
\[
? = \frac{19 \cdot z(z - 7)(z - 5)}{z(z - 7)}
\]
Since \( z(z - 7) \) in the numerator and the denominator cancel out, we have:
\[
? = 19(z - 5)
\]
Thus, the final answer is:
\[
? = 19z - 95
\]
So, the rational expression with the indicated denominator is:
\[
\frac{19(z - 5)}{z(z - 7)(z - 5)} = \frac{19z - 95}{z(z - 7)(z - 5)}
\]
We can factor \( z^2 - 7z \) as follows:
\[
z^2 - 7z = z(z - 7)
\]
Now we set up the equation:
\[
\frac{19}{z(z - 7)} = \frac{?}{z(z - 7)(z - 5)}
\]
To find the numerator represented by the question mark, we can cross multiply:
\[
19 \cdot (z(z - 7)(z - 5)) = ? \cdot (z(z - 7))
\]
This means we need to express \( 19(z(z - 7)(z - 5)) \) divided by \( z(z - 7) \):
\[
? = \frac{19 \cdot z(z - 7)(z - 5)}{z(z - 7)}
\]
Since \( z(z - 7) \) in the numerator and the denominator cancel out, we have:
\[
? = 19(z - 5)
\]
Thus, the final answer is:
\[
? = 19z - 95
\]
So, the rational expression with the indicated denominator is:
\[
\frac{19(z - 5)}{z(z - 7)(z - 5)} = \frac{19z - 95}{z(z - 7)(z - 5)}
\]