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

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

Step 1

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

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

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

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

Cancel the common factor of $x$ .

Factor $x$ out of $x y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $y$ .

Move the leading negative in $- \frac{1}{y}$ into the numerator.

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

Factor $y$ out of $x y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x$ .

Factor $x$ out of $x y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $y$ .

Factor $y$ out of $x y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Simplify the numerator.

Move $- 1$ to the left of $x$ .

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

Rewrite $- 1 x$ as $- x$ .

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

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

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