Questions: Solve the equation x/(6x-36)-5=1/(x-6). Does the equation have a solution? Input Yes or No here: If your answer is Yes, input your solution here: x=

Solution Steps

To solve the equation \(\frac{x}{6x-36} - 5 = \frac{1}{x-6}\), we need to find a common denominator and simplify the equation. We will then solve for \(x\) and check if the solution is valid by substituting it back into the original equation.

Given the equation: \[ \frac{x}{6x - 36} - 5 = \frac{1}{x - 6} \] we can simplify the denominator on the left-hand side: \[ 6x - 36 = 6(x - 6) \] Thus, the equation becomes: \[ \frac{x}{6(x - 6)} - 5 = \frac{1}{x - 6} \]

To combine the terms, we find a common denominator: \[ \frac{x}{6(x - 6)} - \frac{5(x - 6)}{x - 6} = \frac{1}{x - 6} \] This simplifies to: \[ \frac{x - 5 \cdot 6(x - 6)}{6(x - 6)} = \frac{1}{x - 6} \] \[ \frac{x - 30(x - 6)}{6(x - 6)} = \frac{1}{x - 6} \] \[ \frac{x - 30x + 180}{6(x - 6)} = \frac{1}{x - 6} \] \[ \frac{-29x + 180}{6(x - 6)} = \frac{1}{x - 6} \]

Cross-multiplying to eliminate the denominators: \[ -29x + 180 = 6 \] Solving for \(x\): \[ -29x + 180 = 6 \] \[ -29x = 6 - 180 \] \[ -29x = -174 \] \[ x = \frac{174}{29} \] \[ x = 6 \]

We need to check if \(x = 6\) is a valid solution by substituting it back into the original equation: \[ \frac{6}{6 \cdot 6 - 36} - 5 = \frac{1}{6 - 6} \] \[ \frac{6}{36 - 36} - 5 = \frac{1}{0} \] Since division by zero is undefined, \(x = 6\) is not a valid solution.

Final Answer