Solve for z: $\frac{z+9}{z^{2}-81}=1$

Step 1 ：Start with the equation \(\frac{z+9}{z^{2}-81}=1\).

Step 2 ：Multiply both sides of the equation by \(z^{2}-81\) to get rid of the denominator on the left side, resulting in \(z^{2}-81=z+9\).

Step 3 ：Subtract \(z\) and 9 from both sides to get \(z^{2}-z-90=0\).

Step 4 ：Factor the equation to get \((z-10)(z+9)=0\).

Step 5 ：Set each factor equal to zero and solve for \(z\), resulting in \(z=10\) and \(z=-9\).

Step 6 ：The solutions to the equation are \(\boxed{-9}\) and \(\boxed{10}\).