To solve the equation
\[ \frac{9(8 - z)}{8} = -z, \]
first, we can eliminate the fraction by multiplying both sides by 8:
\[ 9(8 - z) = -8z. \]
Next, distribute the 9 on the left side:
\[ 72 - 9z = -8z. \]
Now, we can isolate \(z\). First, add \(9z\) to both sides:
\[ 72 = -8z + 9z. \]
This simplifies to:
\[ 72 = z. \]
Therefore, the solution is
\[ \boxed{72}. \]