Algebra Examples

$\frac{1}{x^{2} - x y} - \frac{y^{2}}{x^{2} - y^{2}} \cdot \frac{x + y}{y^{3}}$

Step 1

Simplify each term.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

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

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Step 1.1.1.1

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

$\frac{1}{x \cdot x - x y} - \frac{y^{2}}{x^{2} - y^{2}} \cdot \frac{x + y}{y^{3}}$

Step 1.1.1.2

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

$\frac{1}{x \cdot x + x (- 1 y)} - \frac{y^{2}}{x^{2} - y^{2}} \cdot \frac{x + y}{y^{3}}$

Step 1.1.1.3

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

$\frac{1}{x (x - 1 y)} - \frac{y^{2}}{x^{2} - y^{2}} \cdot \frac{x + y}{y^{3}}$

Step 1.1.2

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

$\frac{1}{x (x - y)} - \frac{y^{2}}{x^{2} - y^{2}} \cdot \frac{x + y}{y^{3}}$

Step 1.2

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 $b = y$ .

$\frac{1}{x (x - y)} - \frac{y^{2}}{(x + y) (x - y)} \cdot \frac{x + y}{y^{3}}$

Step 1.3

Cancel the common factor of $y^{2}$ .

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Step 1.3.1

Move the leading negative in $- \frac{y^{2}}{(x + y) (x - y)}$ into the numerator.

$\frac{1}{x (x - y)} + \frac{- y^{2}}{(x + y) (x - y)} \cdot \frac{x + y}{y^{3}}$

Step 1.3.2

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

$\frac{1}{x (x - y)} + \frac{y^{2} \cdot - 1}{(x + y) (x - y)} \cdot \frac{x + y}{y^{3}}$

Step 1.3.3

Factor $y^{2}$ out of $y^{3}$ .

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

Step 1.3.4

Cancel the common factor.

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

Step 1.3.5

Rewrite the expression.

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

Step 1.4

Cancel the common factor of $x + y$ .

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Step 1.4.1

Cancel the common factor.

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

Step 1.4.2

Rewrite the expression.

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

Step 1.5

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

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

Step 1.6

Move the negative in front of the fraction.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

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

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Step 6.1

Factor $- 1$ out of $y$ .

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

Reorder terms.

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

Step 6.5

Cancel the common factor.

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

Step 6.6

Rewrite the expression.

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

Step 7

Move the negative in front of the fraction.

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