Solve by Factoring 1/(x-5)-5/(x^2-5x)=3

$\frac{1}{x - 5} - \frac{5}{x^{2} - 5 x} = 3$

Step 1

Subtract $3$ from both sides of the equation.

$\frac{1}{x - 5} - \frac{5}{x^{2} - 5 x} - 3 = 0$

Step 2

Simplify $\frac{1}{x - 5} - \frac{5}{x^{2} - 5 x} - 3$ .

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

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

$\frac{1}{x - 5} - \frac{5}{x \cdot x - 5 x} - 3 = 0$

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

$\frac{1}{x - 5} - \frac{5}{x \cdot x + x \cdot - 5} - 3 = 0$

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

$\frac{1}{x - 5} - \frac{5}{x (x - 5)} - 3 = 0$

To write $\frac{1}{x - 5}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{1}{x - 5} \cdot \frac{x}{x} - \frac{5}{x (x - 5)} - 3 = 0$

Write each expression with a common denominator of $(x - 5) x$ , by multiplying each by an appropriate factor of $1$ .

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

$\frac{x}{(x - 5) x} - \frac{5}{x (x - 5)} - 3 = 0$

Reorder the factors of $(x - 5) x$ .

$\frac{x}{x (x - 5)} - \frac{5}{x (x - 5)} - 3 = 0$

Combine the numerators over the common denominator.

$\frac{x - 5}{x (x - 5)} - 3 = 0$

Cancel the common factor of $x - 5$ .

Cancel the common factor.

$\frac{x - 5}{x (x - 5)} - 3 = 0$

Rewrite the expression.

$\frac{1}{x} - 3 = 0$

Step 3

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{1}{x} - 3 \cdot \frac{x}{x} = 0$

Step 4

Combine $- 3$ and $\frac{x}{x}$ .

$\frac{1}{x} + \frac{- 3 x}{x} = 0$

Step 5

Combine the numerators over the common denominator.

$\frac{1 - 3 x}{x} = 0$

Step 6

Set the numerator equal to zero.

$1 - 3 x = 0$

Step 7

Solve the equation for $x$ .

Subtract $1$ from both sides of the equation.

$- 3 x = - 1$

Divide each term in $- 3 x = - 1$ by $- 3$ and simplify.

Divide each term in $- 3 x = - 1$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{- 1}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{- 1}{- 3}$

Divide $x$ by $1$ .

$x = \frac{- 1}{- 3}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x = \frac{1}{3}$

Solve by Factoring 1/(x-5)-5/(x^2-5x)=3

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