7z+10/z-9=0

To solve the equation \( \frac{7z + 10}{z - 9} = 0 \), we need to set the numerator equal to zero, because a fraction is equal to zero when its numerator is zero (and the denominator is not zero).

1. Set the numerator equal to zero: \[ 7z + 10 = 0 \]

2. Solve for \( z \): \[ 7z = -10 \] \[ z = -\frac{10}{7} \]

Now, we need to ensure that the denominator is not zero for this value: \[ z - 9 \neq 0 \implies z \neq 9 \] Since \( -\frac{10}{7} \) is not equal to \( 9 \), it is a valid solution.

Therefore, the solution to the equation \( \frac{7z + 10}{z - 9} = 0 \) is: \[ z = -\frac{10}{7} \]