Solve by Factoring x/(x-5)=5/(x-5)+6

$\frac{x}{x - 5} = \frac{5}{x - 5} + 6$

Step 1

Move all the expressions to the left side of the equation.

Subtract $\frac{5}{x - 5}$ from both sides of the equation.

$\frac{x}{x - 5} - \frac{5}{x - 5} = 6$

Subtract $6$ from both sides of the equation.

$\frac{x}{x - 5} - \frac{5}{x - 5} - 6 = 0$

Step 2

Simplify $\frac{x}{x - 5} - \frac{5}{x - 5} - 6$ .

Find the common denominator.

Write $- 6$ as a fraction with denominator $1$ .

$\frac{x}{x - 5} - \frac{5}{x - 5} + \frac{- 6}{1} = 0$

Multiply $\frac{- 6}{1}$ by $\frac{x - 5}{x - 5}$ .

$\frac{x}{x - 5} - \frac{5}{x - 5} + \frac{- 6}{1} \cdot \frac{x - 5}{x - 5} = 0$

Multiply $\frac{- 6}{1}$ by $\frac{x - 5}{x - 5}$ .

$\frac{x}{x - 5} - \frac{5}{x - 5} + \frac{- 6 (x - 5)}{x - 5} = 0$

Combine the numerators over the common denominator.

$\frac{x - 5 - 6 (x - 5)}{x - 5} = 0$

Simplify each term.

Apply the distributive property.

$\frac{x - 5 - 6 x - 6 \cdot - 5}{x - 5} = 0$

Multiply $- 6$ by $- 5$ .

$\frac{x - 5 - 6 x + 30}{x - 5} = 0$

Subtract $6 x$ from $x$ .

$\frac{- 5 x - 5 + 30}{x - 5} = 0$

Add $- 5$ and $30$ .

$\frac{- 5 x + 25}{x - 5} = 0$

Factor $5$ out of $- 5 x + 25$ .

Factor $5$ out of $- 5 x$ .

$\frac{5 (- x) + 25}{x - 5} = 0$

Factor $5$ out of $25$ .

$\frac{5 (- x) + 5 (5)}{x - 5} = 0$

Factor $5$ out of $5 (- x) + 5 (5)$ .

$\frac{5 (- x + 5)}{x - 5} = 0$

Cancel the common factor of $- x + 5$ and $x - 5$ .

Factor $- 1$ out of $- x$ .

$\frac{5 (- (x) + 5)}{x - 5} = 0$

Rewrite $5$ as $- 1 (- 5)$ .

$\frac{5 (- (x) - 1 (- 5))}{x - 5} = 0$

Factor $- 1$ out of $- (x) - 1 (- 5)$ .

$\frac{5 (- (x - 5))}{x - 5} = 0$

Rewrite $- (x - 5)$ as $- 1 (x - 5)$ .

$\frac{5 (- 1 (x - 5))}{x - 5} = 0$

Cancel the common factor.

$\frac{5 (- 1 (x - 5))}{x - 5} = 0$

Divide $5 \cdot (- 1)$ by $1$ .

$5 \cdot (- 1) = 0$

Multiply $5$ by $- 1$ .

$- 5 = 0$

Step 3

Since $- 5 \neq 0$ , there are no solutions.

No solution

Solve by Factoring x/(x-5)=5/(x-5)+6

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