Algebra Examples

$\frac{6 x y}{x^{2} - y^{2}} - \frac{x + y}{x - y}$

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 $b = y$ .

$\frac{6 x y}{(x + y) (x - y)} - \frac{x + y}{x - y}$

Step 2

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

$\frac{6 x y}{(x + y) (x - y)} - \frac{x + y}{x - y} \cdot \frac{x + y}{x + y}$

Step 3

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 3.1

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

$\frac{6 x y}{(x + y) (x - y)} - \frac{(x + y) (x + y)}{(x - y) (x + y)}$

Step 3.2

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

$\frac{6 x y}{(x + y) (x - y)} - \frac{(x + y) (x + y)}{(x + y) (x - y)}$

Step 4

Combine the numerators over the common denominator.

$\frac{6 x y - (x + y) (x + y)}{(x + y) (x - y)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{6 x y + (- x - y) (x + y)}{(x + y) (x - y)}$

Step 5.2

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

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Step 5.2.1

Apply the distributive property.

$\frac{6 x y - x (x + y) - y (x + y)}{(x + y) (x - y)}$

Step 5.2.2

Apply the distributive property.

$\frac{6 x y - x \cdot x - x y - y (x + y)}{(x + y) (x - y)}$

Step 5.2.3

Apply the distributive property.

$\frac{6 x y - x \cdot x - x y - y x - y \cdot y}{(x + y) (x - y)}$

Step 5.3

Simplify and combine like terms.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

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

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Step 5.3.1.1.1

Move $x$ .

$\frac{6 x y - (x \cdot x) - x y - y x - y \cdot y}{(x + y) (x - y)}$

Step 5.3.1.1.2

Multiply $x$ by $x$ .

$\frac{6 x y - x^{2} - x y - y x - y \cdot y}{(x + y) (x - y)}$

Step 5.3.1.2

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

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Step 5.3.1.2.1

Move $y$ .

$\frac{6 x y - x^{2} - x y - y x - (y \cdot y)}{(x + y) (x - y)}$

Step 5.3.1.2.2

Multiply $y$ by $y$ .

$\frac{6 x y - x^{2} - x y - y x - y^{2}}{(x + y) (x - y)}$

Step 5.3.2

Subtract $y x$ from $- x y$ .

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Step 5.3.2.1

Move $y$ .

$\frac{6 x y - x^{2} - x y - 1 x y - y^{2}}{(x + y) (x - y)}$

Step 5.3.2.2

Subtract $x y$ from $- x y$ .

$\frac{6 x y - x^{2} - 2 x y - y^{2}}{(x + y) (x - y)}$

Step 5.4

Subtract $2 x y$ from $6 x y$ .

$\frac{- x^{2} + 4 x y - y^{2}}{(x + y) (x - y)}$

Step 6

Simplify with factoring out.

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Step 6.1

Factor $- 1$ out of $- x^{2}$ .

$\frac{- (x^{2}) + 4 x y - y^{2}}{(x + y) (x - y)}$

Step 6.2

Factor $- 1$ out of $4 x y$ .

$\frac{- (x^{2}) - (- 4 x y) - y^{2}}{(x + y) (x - y)}$

Step 6.3

Factor $- 1$ out of $- (x^{2}) - (- 4 x y)$ .

$\frac{- (x^{2} - 4 x y) - y^{2}}{(x + y) (x - y)}$

Step 6.4

Factor $- 1$ out of $- y^{2}$ .

$\frac{- (x^{2} - 4 x y) - (y^{2})}{(x + y) (x - y)}$

Step 6.5

Factor $- 1$ out of $- (x^{2} - 4 x y) - (y^{2})$ .

$\frac{- (x^{2} - 4 x y + y^{2})}{(x + y) (x - y)}$

Step 6.6

Simplify the expression.

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Step 6.6.1

Rewrite $- (x^{2} - 4 x y + y^{2})$ as $- 1 (x^{2} - 4 x y + y^{2})$ .

$\frac{- 1 (x^{2} - 4 x y + y^{2})}{(x + y) (x - y)}$

Step 6.6.2

Move the negative in front of the fraction.

$- \frac{x^{2} - 4 x y + y^{2}}{(x + y) (x - y)}$