1Cademy - Example 7.26: Simplifying $$\frac{\frac{1}{x}+\frac{1}{y}}{\frac{x}{y}-\frac{y}{x}}$$ by Writing it as Division

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Simplifying Complex Fractions

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Example 7.26: Simplifying $\frac{\frac{1}{x}+\frac{1}{y}}{\frac{x}{y}-\frac{y}{x}}$ by Writing it as Division

To simplify the complex rational expression $\frac{\frac{1}{x}+\frac{1}{y}}{\frac{x}{y}-\frac{y}{x}}$ using division, apply the following procedure:

Step 1 — Simplify the numerator and denominator. In the numerator, the common denominator is $xy$ . Multiplying $\frac{1}{x}$ by $\frac{y}{y}$ and $\frac{1}{y}$ by $\frac{x}{x}$ yields $\frac{y}{xy} + \frac{x}{xy} = \frac{y + x}{xy}$ . In the denominator, the common denominator is also $xy$ . Multiplying $\frac{x}{y}$ by $\frac{x}{x}$ and $\frac{y}{x}$ by $\frac{y}{y}$ gives $\frac{x^2}{xy} - \frac{y^2}{xy} = \frac{x^2 - y^2}{xy}$ . The expression simplifies to $\frac{\frac{y + x}{xy}}{\frac{x^2 - y^2}{xy}}$ .

Step 2 — Rewrite as a division problem. The main fraction bar indicates division, so rewrite it as $\frac{y + x}{xy} \div \frac{x^2 - y^2}{xy}$ .

Step 3 — Multiply the first expression by the reciprocal of the second. Change the division to multiplication: $\frac{y + x}{xy} \cdot \frac{xy}{x^2 - y^2}$ .

Step 4 — Simplify. Factor the difference of squares in the denominator to $(x - y)(x + y)$ . The expression becomes $\frac{xy(y + x)}{xy(x - y)(x + y)}$ . Removing the common factors $xy$ and $(x + y)$ (which is equivalent to $(y + x)$ ) leaves $\frac{1}{x - y}$ .

Updated 2026-04-30

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OpenStax Intermediate Algebra

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