Solve by Factoring x/(x-4)=4/(x-4)+5

$\frac{x}{x - 4} = \frac{4}{x - 4} + 5$

Step 1

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

Subtract $\frac{4}{x - 4}$ from both sides of the equation.

$\frac{x}{x - 4} - \frac{4}{x - 4} = 5$

Subtract $5$ from both sides of the equation.

$\frac{x}{x - 4} - \frac{4}{x - 4} - 5 = 0$

Step 2

Simplify $\frac{x}{x - 4} - \frac{4}{x - 4} - 5$ .

Find the common denominator.

Write $- 5$ as a fraction with denominator $1$ .

$\frac{x}{x - 4} - \frac{4}{x - 4} + \frac{- 5}{1} = 0$

Multiply $\frac{- 5}{1}$ by $\frac{x - 4}{x - 4}$ .

$\frac{x}{x - 4} - \frac{4}{x - 4} + \frac{- 5}{1} \cdot \frac{x - 4}{x - 4} = 0$

Multiply $\frac{- 5}{1}$ by $\frac{x - 4}{x - 4}$ .

$\frac{x}{x - 4} - \frac{4}{x - 4} + \frac{- 5 (x - 4)}{x - 4} = 0$

Combine the numerators over the common denominator.

$\frac{x - 4 - 5 (x - 4)}{x - 4} = 0$

Simplify each term.

Apply the distributive property.

$\frac{x - 4 - 5 x - 5 \cdot - 4}{x - 4} = 0$

Multiply $- 5$ by $- 4$ .

$\frac{x - 4 - 5 x + 20}{x - 4} = 0$

Subtract $5 x$ from $x$ .

$\frac{- 4 x - 4 + 20}{x - 4} = 0$

Add $- 4$ and $20$ .

$\frac{- 4 x + 16}{x - 4} = 0$

Factor $4$ out of $- 4 x + 16$ .

Factor $4$ out of $- 4 x$ .

$\frac{4 (- x) + 16}{x - 4} = 0$

Factor $4$ out of $16$ .

$\frac{4 (- x) + 4 (4)}{x - 4} = 0$

Factor $4$ out of $4 (- x) + 4 (4)$ .

$\frac{4 (- x + 4)}{x - 4} = 0$

Cancel the common factor of $- x + 4$ and $x - 4$ .

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

$\frac{4 (- (x) + 4)}{x - 4} = 0$

Rewrite $4$ as $- 1 (- 4)$ .

$\frac{4 (- (x) - 1 (- 4))}{x - 4} = 0$

Factor $- 1$ out of $- (x) - 1 (- 4)$ .

$\frac{4 (- (x - 4))}{x - 4} = 0$

Rewrite $- (x - 4)$ as $- 1 (x - 4)$ .

$\frac{4 (- 1 (x - 4))}{x - 4} = 0$

Cancel the common factor.

$\frac{4 (- 1 (x - 4))}{x - 4} = 0$

Divide $4 \cdot (- 1)$ by $1$ .

$4 \cdot (- 1) = 0$

Multiply $4$ by $- 1$ .

$- 4 = 0$

Step 3

Since $- 4 \neq 0$ , there are no solutions.

No solution

Solve by Factoring x/(x-4)=4/(x-4)+5

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