Algebra Examples

$\frac{\frac{x^{2} - y^{2}}{x y}}{\frac{1}{y} - \frac{1}{x}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

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

Step 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 $b = y$ .

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

Step 3

Simplify the denominator.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

Reorder the factors of $y x$ .

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

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

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

Step 6

Cancel the common factor of $x - y$ .

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Step 6.1

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

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

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

Step 7

Cancel the common factor of $x y$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

$x + y$