1Cademy - Simplifying $$\frac{\frac{1}{x}+\frac{1}{y}}{\frac{x}{y}-\frac{y}{x}}$$ Using the LCD

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Simplifying a Complex Rational Expression by Using the LCD

Example

Simplifying $\frac{\frac{1}{x}+\frac{1}{y}}{\frac{x}{y}-\frac{y}{x}}$ Using the LCD

Simplify the complex rational expression using the LCD method:

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

Step 1 — Find the LCD of all inner fractions. The inner denominators are $x$ , $y$ , $y$ , and $x$ . Since $x$ and $y$ are distinct variable factors, the LCD is $xy$ .

Step 2 — Multiply the numerator and denominator of the complex fraction by the LCD $xy$ . Distribute $xy$ to each term:

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

Step 3 — Simplify by canceling matching factors. In the numerator, $xy \cdot \frac{1}{x} = y$ and $xy \cdot \frac{1}{y} = x$ . In the denominator, $xy \cdot \frac{x}{y} = x^2$ and $xy \cdot \frac{y}{x} = y^2$ :

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

Step 4 — Factor the denominator and cancel common factors. Recognize $x^2 - y^2$ as a difference of squares: $x^2 - y^2 = (x + y)(x - y)$ . The numerator $y + x$ is the same as $x + y$ , so:

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

The simplified result is $\frac{1}{x - y}$ . This example demonstrates the LCD method applied to a two-variable complex rational expression where all four inner denominators are single variables. Multiplying every term by the LCD of $xy$ clears all fractions in one step. After simplification, the denominator factors as a difference of squares, revealing a common factor with the numerator that allows the expression to reduce to a simple fraction.

Updated 2026-04-30

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