Solve by Factoring 1/(x^2-5x)=(x+7)/x-1

$\frac{1}{x^{2} - 5 x} = \frac{x + 7}{x} - 1$

Step 1

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

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

$\frac{1}{x^{2} - 5 x} - \frac{x + 7}{x} = - 1$

Add $1$ to both sides of the equation.

$\frac{1}{x^{2} - 5 x} - \frac{x + 7}{x} + 1 = 0$

Step 2

Simplify $\frac{1}{x^{2} - 5 x} - \frac{x + 7}{x} + 1$ .

Factor $x$ out of $x^{2} - 5 x$ .

Factor $x$ out of $x^{2}$ .

$\frac{1}{x \cdot x - 5 x} - \frac{x + 7}{x} + 1 = 0$

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

$\frac{1}{x \cdot x + x \cdot - 5} - \frac{x + 7}{x} + 1 = 0$

Factor $x$ out of $x \cdot x + x \cdot - 5$ .

$\frac{1}{x (x - 5)} - \frac{x + 7}{x} + 1 = 0$

To write $- \frac{x + 7}{x}$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

$\frac{1}{x (x - 5)} - \frac{x + 7}{x} \cdot \frac{x - 5}{x - 5} + 1 = 0$

Multiply $\frac{x + 7}{x}$ by $\frac{x - 5}{x - 5}$ .

$\frac{1}{x (x - 5)} - \frac{(x + 7) (x - 5)}{x (x - 5)} + 1 = 0$

Combine the numerators over the common denominator.

$\frac{1 - (x + 7) (x - 5)}{x (x - 5)} + 1 = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{1 + (- x - 1 \cdot 7) (x - 5)}{x (x - 5)} + 1 = 0$

Multiply $- 1$ by $7$ .

$\frac{1 + (- x - 7) (x - 5)}{x (x - 5)} + 1 = 0$

Expand $(- x - 7) (x - 5)$ using the FOIL Method.

Apply the distributive property.

$\frac{1 - x (x - 5) - 7 (x - 5)}{x (x - 5)} + 1 = 0$

Apply the distributive property.

$\frac{1 - x \cdot x - x \cdot - 5 - 7 (x - 5)}{x (x - 5)} + 1 = 0$

Apply the distributive property.

$\frac{1 - x \cdot x - x \cdot - 5 - 7 x - 7 \cdot - 5}{x (x - 5)} + 1 = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{1 - (x \cdot x) - x \cdot - 5 - 7 x - 7 \cdot - 5}{x (x - 5)} + 1 = 0$

Multiply $x$ by $x$ .

$\frac{1 - x^{2} - x \cdot - 5 - 7 x - 7 \cdot - 5}{x (x - 5)} + 1 = 0$

Multiply $- 5$ by $- 1$ .

$\frac{1 - x^{2} + 5 x - 7 x - 7 \cdot - 5}{x (x - 5)} + 1 = 0$

Multiply $- 7$ by $- 5$ .

$\frac{1 - x^{2} + 5 x - 7 x + 35}{x (x - 5)} + 1 = 0$

Subtract $7 x$ from $5 x$ .

$\frac{1 - x^{2} - 2 x + 35}{x (x - 5)} + 1 = 0$

Add $1$ and $35$ .

$\frac{- x^{2} - 2 x + 36}{x (x - 5)} + 1 = 0$

Write $1$ as a fraction with a common denominator.

$\frac{- x^{2} - 2 x + 36}{x (x - 5)} + \frac{x (x - 5)}{x (x - 5)} = 0$

Combine the numerators over the common denominator.

$\frac{- x^{2} - 2 x + 36 + x (x - 5)}{x (x - 5)} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{- x^{2} - 2 x + 36 + x \cdot x + x \cdot - 5}{x (x - 5)} = 0$

Multiply $x$ by $x$ .

$\frac{- x^{2} - 2 x + 36 + x^{2} + x \cdot - 5}{x (x - 5)} = 0$

Move $- 5$ to the left of $x$ .

$\frac{- x^{2} - 2 x + 36 + x^{2} - 5 \cdot x}{x (x - 5)} = 0$

Add $- x^{2}$ and $x^{2}$ .

$\frac{- 2 x + 36 + 0 - 5 x}{x (x - 5)} = 0$

Add $- 2 x$ and $0$ .

$\frac{- 2 x + 36 - 5 x}{x (x - 5)} = 0$

Subtract $5 x$ from $- 2 x$ .

$\frac{- 7 x + 36}{x (x - 5)} = 0$

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

$\frac{- (7 x) + 36}{x (x - 5)} = 0$

Rewrite $36$ as $- 1 (- 36)$ .

$\frac{- (7 x) - 1 (- 36)}{x (x - 5)} = 0$

Factor $- 1$ out of $- (7 x) - 1 (- 36)$ .

$\frac{- (7 x - 36)}{x (x - 5)} = 0$

Rewrite $- (7 x - 36)$ as $- 1 (7 x - 36)$ .

$\frac{- 1 (7 x - 36)}{x (x - 5)} = 0$

Move the negative in front of the fraction.

$- \frac{7 x - 36}{x (x - 5)} = 0$

Step 3

Set the numerator equal to zero.

$7 x - 36 = 0$

Step 4

Solve the equation for $x$ .

Add $36$ to both sides of the equation.

$7 x = 36$

Divide each term in $7 x = 36$ by $7$ and simplify.

Divide each term in $7 x = 36$ by $7$ .

$\frac{7 x}{7} = \frac{36}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 x}{7} = \frac{36}{7}$

Divide $x$ by $1$ .

$x = \frac{36}{7}$

Solve by Factoring 1/(x^2-5x)=(x+7)/x-1

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