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

1. Problem Description

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

2. Solution Steps

First, we factor the denominator $6x-36$ as $6(x-6)$. The given 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. Note that $x \neq 6$, as this would make the denominators zero.

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

Distribute $6(x-6)$ on the left side:

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

Simplify each term:

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

Expand the term $54(x-6)$:

$x - 54x + 324 = 6$

Combine like terms on the left side:

$-53x + 324 = 6$

Subtract 324 from both sides:

$-53x = 6 - 324$

$-53x = -318$

Divide both sides by $-53$:

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

$x = 6$

However, we noted that $x \neq 6$ because this would make the denominator zero.

Therefore, there is no solution.

3. Final Answer

No solution.