Simplify 5/(x^2-xy)-5/(y^2-xy)

$\frac{5}{x^{2} - x y} - \frac{5}{y^{2} - x y}$

Step 1

Simplify each term.

Simplify the denominator.

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

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

$\frac{5}{x \cdot x - x y} - \frac{5}{y^{2} - x y}$

Factor $x$ out of $- x y$ .

$\frac{5}{x \cdot x + x (- 1 y)} - \frac{5}{y^{2} - x y}$

Factor $x$ out of $x \cdot x + x (- 1 y)$ .

$\frac{5}{x (x - 1 y)} - \frac{5}{y^{2} - x y}$

Rewrite $- 1 y$ as $- y$ .

$\frac{5}{x (x - y)} - \frac{5}{y^{2} - x y}$

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

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

$\frac{5}{x (x - y)} - \frac{5}{y \cdot y - x y}$

Factor $y$ out of $- x y$ .

$\frac{5}{x (x - y)} - \frac{5}{y \cdot y + y (- x)}$

Factor $y$ out of $y \cdot y + y (- x)$ .

$\frac{5}{x (x - y)} - \frac{5}{y (y - x)}$

Step 2

Simplify with factoring out.

Factor $- 1$ out of $y$ .

$\frac{5}{x (x - y)} - \frac{5}{y (- 1 (- y) - x)}$

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

$\frac{5}{x (x - y)} - \frac{5}{y (- 1 (- y) - (x))}$

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

$\frac{5}{x (x - y)} - \frac{5}{y (- 1 (- y + x))}$

Reorder terms.

$\frac{5}{x (x - y)} - \frac{5}{y (- 1 (x - y))}$

Step 3

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

$\frac{5}{x (x - y)} \cdot \frac{y \cdot - 1}{y \cdot - 1} - \frac{5}{y (- 1 (x - y))}$

Step 4

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

$\frac{5}{x (x - y)} \cdot \frac{y \cdot - 1}{y \cdot - 1} - \frac{5}{y (- 1 (x - y))} \cdot \frac{x}{x}$

Step 5

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

Multiply $\frac{5}{x (x - y)}$ by $\frac{y \cdot - 1}{y \cdot - 1}$ .

$\frac{5 (y \cdot - 1)}{x (x - y) (y \cdot - 1)} - \frac{5}{y (- 1 (x - y))} \cdot \frac{x}{x}$

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

$\frac{5 (y \cdot - 1)}{x (x - y) (y \cdot - 1)} - \frac{5 x}{y (- 1 (x - y)) x}$

Reorder the factors of $x (x - y) (y \cdot - 1)$ .

$\frac{5 (y \cdot - 1)}{- x y (x - y)} - \frac{5 x}{y (- 1 (x - y)) x}$

Reorder the factors of $y (- 1 (x - y)) x$ .

$\frac{5 (y \cdot - 1)}{- x y (x - y)} - \frac{5 x}{- x y (x - y)}$

Step 6

Combine the numerators over the common denominator.

$\frac{5 (y \cdot - 1) - 5 x}{- x y (x - y)}$

Step 7

Simplify the numerator.

Factor $5$ out of $5 y \cdot - 1 - 5 x$ .

Factor $5$ out of $5 y \cdot - 1$ .

$\frac{5 (y \cdot - 1) - 5 x}{- x y (x - y)}$

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

$\frac{5 (y \cdot - 1) + 5 (- x)}{- x y (x - y)}$

Factor $5$ out of $5 (y \cdot - 1) + 5 (- x)$ .

$\frac{5 (y \cdot - 1 - x)}{- x y (x - y)}$

Move $- 1$ to the left of $y$ .

$\frac{5 (- 1 \cdot y - x)}{- x y (x - y)}$

Rewrite $- 1 y$ as $- y$ .

$\frac{5 (- y - x)}{- x y (x - y)}$

Step 8

Simplify with factoring out.

Move the negative in front of the fraction.

$- \frac{5 (- y - x)}{x y (x - y)}$

Factor $- 1$ out of $- y$ .

$- \frac{5 (- (y) - x)}{x y (x - y)}$

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

$- \frac{5 (- (y) - (x))}{x y (x - y)}$

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

$- \frac{5 (- (y + x))}{x y (x - y)}$

Simplify the expression.

Rewrite $- (y + x)$ as $- 1 (y + x)$ .

$- \frac{5 (- 1 (y + x))}{x y (x - y)}$

Move the negative in front of the fraction.

$- - \frac{5 (y + x)}{x y (x - y)}$

Multiply $- 1$ by $- 1$ .

$1 \frac{5 (y + x)}{x y (x - y)}$

Multiply $\frac{5 (y + x)}{x y (x - y)}$ by $1$ .

$\frac{5 (y + x)}{x y (x - y)}$

Simplify 5/(x^2-xy)-5/(y^2-xy)

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