Simplify (1/x+1/y)/(x/y-y/x)

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

Step 1

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

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

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

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.

Cancel the common factor of $x$ .

Factor $x$ out of $x y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $y$ .

Factor $y$ out of $x y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $y$ .

Factor $y$ out of $x y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Cancel the common factor of $x$ .

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

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

Factor $x$ out of $x y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

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 $x + y$ .

Reorder terms.

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify (1/x+1/y)/(x/y-y/x)

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