We are given the equation $\frac{x}{6x-36} - 9 = \frac{1}{x-6}$ and we need to solve for $x$.

1. Problem Description

We are given the equation $\frac{x}{6x-36} - 9 = \frac{1}{x-6}$ and we need to solve for $x$.

2. Solution Steps

First, factor the denominator on the left side:

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

So the equation becomes:

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

Multiply both sides of the equation by $6(x-6)$ to eliminate the denominators. We must note that $x \neq 6$, otherwise the denominators would be zero:

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

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

$x - 54x + 324 = 6$

$-53x = 6 - 324$

$-53x = -318$

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

$x = 6$

However, we noted earlier that $x \neq 6$ because it makes the denominators zero. Therefore, there is no solution.

3. Final Answer

No solution.