### Home > PC3 > Chapter 5 > Lesson 5.1.3 > Problem5-31

5-31.

​Rewrite each the following expressions as a single, simplified fraction.

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

1. Multiply out.
2. Simplify.
3. Rewrite using a common denominator.

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

Multiply both the numerator and denominator of the fraction by $\frac{xy}{1}$.
$\frac{y^2-x^2}{y^2+x^2}+1$

$\text{Change the } 1 \text{ to: }\frac{y^2+x^2}{y^2+x^2}$
$\frac{\textit{y}^2-\textit{x}^2}{{\textit{y}^2+\textit{x}^2}}+\frac{\textit{y}^2+\textit{x}^2}{{\textit{y}^2+\textit{x}^2}}$

Combine the two fractions.
$\frac{\textit{y}^2 - \textit{x}^2 + \textit{y}^2 + \textit{x}^2}{\textit{y}^2 + \textit{x}^2}$

Simplify.
$\frac{2\textit{y}^2}{\textit{y}^2+\textit{x}^2}$