Algebra Examples

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

Step 1

Multiply the numerator and denominator of the fraction by $x y$ .

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Step 1.1

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

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x$ .

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Step 3.1.1

Factor $x$ out of $x y$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $y$ .

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Step 3.2.1

Factor $y$ out of $x y$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $y$ .

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Step 3.3.1

Factor $y$ out of $x y$ .

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 3.4

Raise $x$ to the power of $1$ .

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

Step 3.5

Raise $x$ to the power of $1$ .

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

Step 3.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.7

Add $1$ and $1$ .

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

Step 3.8

Cancel the common factor of $x$ .

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Step 3.8.1

Move the leading negative in $- \frac{y}{x}$ into the numerator.

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

Step 3.8.2

Factor $x$ out of $x y$ .

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

Step 3.8.3

Cancel the common factor.

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

Step 3.8.4

Rewrite the expression.

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

Step 3.9

Raise $y$ to the power of $1$ .

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

Step 3.10

Raise $y$ to the power of $1$ .

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

Step 3.11

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.12

Add $1$ and $1$ .

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

Step 4

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 $b = y$ .

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

Step 5

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

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Step 5.1

Reorder terms.

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

Step 5.2

Cancel the common factor.

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

Step 5.3

Rewrite the expression.

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