Algebra Examples

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

Step 1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = y$ .

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

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

Simplify terms.

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Step 5.1

Combine the numerators over the common denominator.

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

Step 5.2

Combine the numerators over the common denominator.

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

Step 6

Simplify each term.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $x$ by $x$ .

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

Step 6.3

Apply the distributive property.

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

Step 6.4

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

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Step 6.4.1

Move $x$ .

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

Step 6.4.2

Multiply $x$ by $x$ .

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

Step 6.5

Rewrite using the commutative property of multiplication.

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

Step 6.6

Multiply $- 2$ by $- 1$ .

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

Step 7

Simplify terms.

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Step 7.1

Combine the opposite terms in $x^{2} + x y - 2 x^{2} + 2 x y - 2 x y$ .

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Step 7.1.1

Subtract $2 x y$ from $2 x y$ .

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

Step 7.1.2

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

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

Step 7.2

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

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

Step 7.3

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

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Step 7.3.1

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

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

Step 7.3.2

Factor $x$ out of $x y$ .

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

Step 7.3.3

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

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

Step 7.4

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

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Step 7.4.1

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

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

Step 7.4.2

Factor $- 1$ out of $y$ .

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

Step 7.4.3

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

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

Step 7.4.4

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

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

Step 7.4.5

Cancel the common factor.

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

Step 7.4.6

Rewrite the expression.

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

Step 7.5

Simplify the expression.

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Step 7.5.1

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

$\frac{- 1 \cdot x}{x + y}$

Step 7.5.2

Move the negative in front of the fraction.

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