Question

Simplify the complex rational expression by writing it as division:

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

Short answer

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

Step-by-step solution

Step 1. Given information.

The following complex rational expression is given:

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

Step 2. Simplification of the complex rational expression.

Find a common denominator and add the fractions present at the numerator:

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

Find a common denominator and subtract the terms present at the denominator:

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

Simplify:

$$\begin{gathered} \frac{{\displaystyle \frac{y}{xy}}+{\displaystyle \frac{x}{xy}}}{{\displaystyle \frac{y}{xy}}-{\displaystyle \frac{x}{xy}}} \\ \frac{{\displaystyle \frac{y+x}{xy}}}{{\displaystyle \frac{y-x}{xy}}} \end{gathered}$$

Write as a division problem:

$\frac{y+x}{xy}÷\frac{y-x}{xy}$

Write as the product of the first and the reciprocal of the second:

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

Cancel out the common factors from the numerator and the denominator and simplify:

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