We are asked to solve the equation $\frac{10}{z^2 - 25} = \frac{5}{z^2 - 5z}$ and determine if there is a solution or not.

1. Problem Description

We are asked to solve the equation $\frac{10}{z^2 - 25} = \frac{5}{z^2 - 5z}$ and determine if there is a solution or not.

2. Solution Steps

First, we can cross-multiply the equation:

$10(z^2 - 5z) = 5(z^2 - 25)$

Now, we can simplify the equation:

$10z^2 - 50z = 5z^2 - 125$

Subtract $5z^2$ from both sides:

$5z^2 - 50z = -125$

Add $125$ to both sides:

$5z^2 - 50z + 125 = 0$

Divide the equation by 5:

$z^2 - 10z + 25 = 0$

Now, we can factor the quadratic equation:

$(z - 5)(z - 5) = 0$

$(z - 5)^2 = 0$

Solve for $z$:

$z - 5 = 0$

$z = 5$

However, we need to check for extraneous solutions.

If $z = 5$, then the denominators of the original equation are:

$z^2 - 25 = 5^2 - 25 = 25 - 25 = 0$

$z^2 - 5z = 5^2 - 5(5) = 25 - 25 = 0$

Since both denominators become 0 when $z = 5$, the solution $z = 5$ is an extraneous solution. Therefore, there is no solution to the equation.

3. Final Answer

B. There is no solution.