Pre-Algebra Examples

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

Step 1

Simplify the denominator.

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

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

$\frac{x + y}{x \cdot x - x y} - \frac{x - y}{x y} - \frac{2}{x - y}$

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.2

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

$\frac{x + y}{x (x - y)} - \frac{x - y}{x y} - \frac{2}{x - y}$

Step 2

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

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

Step 3

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

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

Step 4

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

Tap for more steps...

Step 4.1

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

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

Step 4.2

Multiply $\frac{x - y}{x y}$ by $\frac{x - y}{x - y}$ .

$\frac{(x + y) y}{x (x - y) y} - \frac{(x - y) (x - y)}{x y (x - y)} - \frac{2}{x - y}$

Step 4.3

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

$\frac{(x + y) y}{x y (x - y)} - \frac{(x - y) (x - y)}{x y (x - y)} - \frac{2}{x - y}$

Step 5

Combine the numerators over the common denominator.

$\frac{(x + y) y - (x - y) (x - y)}{x y (x - y)} - \frac{2}{x - y}$

Step 6

Simplify each term.

Tap for more steps...

Step 6.1

Simplify the numerator.

Tap for more steps...

Step 6.1.1

Apply the distributive property.

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

Step 6.1.2

Multiply $y$ by $y$ .

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

Step 6.1.3

Apply the distributive property.

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

Step 6.1.4

Multiply $- - y$ .

Tap for more steps...

Step 6.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 6.1.4.2

Multiply $y$ by $1$ .

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

Step 6.1.5

Expand $(- x + y) (x - y)$ using the FOIL Method.

Tap for more steps...

Step 6.1.5.1

Apply the distributive property.

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

Step 6.1.5.2

Apply the distributive property.

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

Step 6.1.5.3

Apply the distributive property.

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

Step 6.1.6

Simplify and combine like terms.

Tap for more steps...

Step 6.1.6.1

Simplify each term.

Tap for more steps...

Step 6.1.6.1.1

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

Tap for more steps...

Step 6.1.6.1.1.1

Move $x$ .

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

Step 6.1.6.1.1.2

Multiply $x$ by $x$ .

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

Step 6.1.6.1.2

Rewrite using the commutative property of multiplication.

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

Step 6.1.6.1.3

Multiply $- 1$ by $- 1$ .

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

Step 6.1.6.1.4

Multiply $x$ by $1$ .

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

Step 6.1.6.1.5

Rewrite using the commutative property of multiplication.

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

Step 6.1.6.1.6

Multiply $y$ by $y$ by adding the exponents.

Tap for more steps...

Step 6.1.6.1.6.1

Move $y$ .

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

Step 6.1.6.1.6.2

Multiply $y$ by $y$ .

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

Step 6.1.6.2

Add $x y$ and $y x$ .

Tap for more steps...

Step 6.1.6.2.1

Reorder $y$ and $x$ .

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

Step 6.1.6.2.2

Add $x y$ and $x y$ .

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

Step 6.1.7

Add $x y$ and $2 x y$ .

$\frac{y^{2} - x^{2} + 3 x y - y^{2}}{x y (x - y)} - \frac{2}{x - y}$

Step 6.1.8

Subtract $y^{2}$ from $y^{2}$ .

$\frac{- x^{2} + 3 x y + 0}{x y (x - y)} - \frac{2}{x - y}$

Step 6.1.9

Add $- x^{2} + 3 x y$ and $0$ .

$\frac{- x^{2} + 3 x y}{x y (x - y)} - \frac{2}{x - y}$

Step 6.1.10

Factor $x$ out of $- x^{2} + 3 x y$ .

Tap for more steps...

Step 6.1.10.1

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

$\frac{x (- x) + 3 x y}{x y (x - y)} - \frac{2}{x - y}$

Step 6.1.10.2

Factor $x$ out of $3 x y$ .

$\frac{x (- x) + x (3 y)}{x y (x - y)} - \frac{2}{x - y}$

Step 6.1.10.3

Factor $x$ out of $x (- x) + x (3 y)$ .

$\frac{x (- x + 3 y)}{x y (x - y)} - \frac{2}{x - y}$

Step 6.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 6.2.1

Cancel the common factor.

$\frac{x (- x + 3 y)}{x y (x - y)} - \frac{2}{x - y}$

Step 6.2.2

Rewrite the expression.

$\frac{- x + 3 y}{y (x - y)} - \frac{2}{x - y}$

Step 7

To write $- \frac{2}{x - y}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$\frac{- x + 3 y}{y (x - y)} - \frac{2}{x - y} \cdot \frac{y}{y}$

Step 8

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

Tap for more steps...

Step 8.1

Multiply $\frac{2}{x - y}$ by $\frac{y}{y}$ .

$\frac{- x + 3 y}{y (x - y)} - \frac{2 y}{(x - y) y}$

Step 8.2

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

$\frac{- x + 3 y}{y (x - y)} - \frac{2 y}{y (x - y)}$

Step 9

Combine the numerators over the common denominator.

$\frac{- x + 3 y - 2 y}{y (x - y)}$

Step 10

Subtract $2 y$ from $3 y$ .

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

Step 11

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

Tap for more steps...

Step 11.1

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

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

Step 11.2

Factor $- 1$ out of $y$ .

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

Step 11.3

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

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

Step 11.4

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

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

Step 11.5

Cancel the common factor.

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

Step 11.6

Rewrite the expression.

$\frac{- 1}{y}$

Step 12

Move the negative in front of the fraction.

$- \frac{1}{y}$