To complete the expression
\[ \frac{19}{z^2 - 7z} = \frac{ \text{nothing} }{ z (z - 7)(z - 5) } \]
we first need to factor the denominator on the left side.
The expression \( z^2 - 7z \) can be factored as follows:
\[ z^2 - 7z = z(z - 7) \]
So we rewrite the left side of the equation:
\[ \frac{19}{z(z - 7)} = \frac{\text{nothing}}{z(z - 7)(z - 5)} \]
To have a common denominator on both sides of the equation, we can multiply the numerator on the left side by \((z - 5)\):
\[ \frac{19(z - 5)}{z(z - 7)(z - 5)} = \frac{\text{nothing}}{z(z - 7)(z - 5)} \]
Now, we can see that the numerator on the left side is:
\[ 19(z - 5) \]
So the completed expression is:
\[ \frac{19(z - 5)}{z(z - 7)(z - 5)} \]
To answer the original question, we can say:
\[ \text{nothing} = 19(z - 5) \]
Thus, the final expression is:
\[ \frac{19(z - 5)}{z(z - 7)(z - 5)} \]
In this expression, you can simplify it:
\[ \frac{19}{z(z - 7)} \quad \text{for } z \neq 5 \]
is the simplified form.