Solve by Factoring x/(x-7)=7/(x-7)+8

$\frac{x}{x - 7} = \frac{7}{x - 7} + 8$

Step 1

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

Subtract $\frac{7}{x - 7}$ from both sides of the equation.

$\frac{x}{x - 7} - \frac{7}{x - 7} = 8$

Subtract $8$ from both sides of the equation.

$\frac{x}{x - 7} - \frac{7}{x - 7} - 8 = 0$

Step 2

Simplify $\frac{x}{x - 7} - \frac{7}{x - 7} - 8$ .

Find the common denominator.

Write $- 8$ as a fraction with denominator $1$ .

$\frac{x}{x - 7} - \frac{7}{x - 7} + \frac{- 8}{1} = 0$

Multiply $\frac{- 8}{1}$ by $\frac{x - 7}{x - 7}$ .

$\frac{x}{x - 7} - \frac{7}{x - 7} + \frac{- 8}{1} \cdot \frac{x - 7}{x - 7} = 0$

Multiply $\frac{- 8}{1}$ by $\frac{x - 7}{x - 7}$ .

$\frac{x}{x - 7} - \frac{7}{x - 7} + \frac{- 8 (x - 7)}{x - 7} = 0$

Combine the numerators over the common denominator.

$\frac{x - 7 - 8 (x - 7)}{x - 7} = 0$

Simplify each term.

Apply the distributive property.

$\frac{x - 7 - 8 x - 8 \cdot - 7}{x - 7} = 0$

Multiply $- 8$ by $- 7$ .

$\frac{x - 7 - 8 x + 56}{x - 7} = 0$

Subtract $8 x$ from $x$ .

$\frac{- 7 x - 7 + 56}{x - 7} = 0$

Add $- 7$ and $56$ .

$\frac{- 7 x + 49}{x - 7} = 0$

Factor $7$ out of $- 7 x + 49$ .

Factor $7$ out of $- 7 x$ .

$\frac{7 (- x) + 49}{x - 7} = 0$

Factor $7$ out of $49$ .

$\frac{7 (- x) + 7 (7)}{x - 7} = 0$

Factor $7$ out of $7 (- x) + 7 (7)$ .

$\frac{7 (- x + 7)}{x - 7} = 0$

Cancel the common factor of $- x + 7$ and $x - 7$ .

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

$\frac{7 (- (x) + 7)}{x - 7} = 0$

Rewrite $7$ as $- 1 (- 7)$ .

$\frac{7 (- (x) - 1 (- 7))}{x - 7} = 0$

Factor $- 1$ out of $- (x) - 1 (- 7)$ .

$\frac{7 (- (x - 7))}{x - 7} = 0$

Rewrite $- (x - 7)$ as $- 1 (x - 7)$ .

$\frac{7 (- 1 (x - 7))}{x - 7} = 0$

Cancel the common factor.

$\frac{7 (- 1 (x - 7))}{x - 7} = 0$

Divide $7 \cdot (- 1)$ by $1$ .

$7 \cdot (- 1) = 0$

Multiply $7$ by $- 1$ .

$- 7 = 0$

Step 3

Since $- 7 \neq 0$ , there are no solutions.

No solution

Solve by Factoring x/(x-7)=7/(x-7)+8

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