The problem is to solve the equation $\frac{x}{6x-36} - 9 = \frac{1}{x-6}$ for $x$.

1. Problem Description

The problem is to solve the equation $\frac{x}{6x-36} - 9 = \frac{1}{x-6}$ for $x$.

2. Solution Steps

First, we factor the denominator $6x-36$:

$6x - 36 = 6(x-6)$.

Now, rewrite the equation:

$\frac{x}{6(x-6)} - 9 = \frac{1}{x-6}$.

Multiply both sides of the equation by $6(x-6)$ to eliminate the denominators:

$6(x-6)\left(\frac{x}{6(x-6)} - 9\right) = 6(x-6)\left(\frac{1}{x-6}\right)$

$x - 54(x-6) = 6$

$x - 54x + 324 = 6$

$-53x + 324 = 6$

$-53x = 6 - 324$

$-53x = -318$

$x = \frac{-318}{-53}$

$x = 6$.

However, we must check if this solution is valid. The original equation has terms with denominators $6x-36 = 6(x-6)$ and $x-6$. If $x=6$, then both denominators are zero, which means the equation is undefined. Therefore, $x=6$ is not a valid solution.

Thus, there is no solution to this equation.

3. Final Answer

No solution.