Chapter 6.5, Problem 40WE

To determine

Find the simplified form of the given expression.

Answer to Problem 40WE

The simplified form of the expression isa $\frac{x^2+y^2}{\left(x+y\right)\left(x-y\right)}$ .

Explanation of Solution

Given:

The given expression is $\frac{x}{x+y}+\frac{y}{x-y}$ .

Calculation:

Since the denominators are different, rewrite the fractions using their least common denominator, $\left(x+y\right)\left(x-y\right)$ .

  $\begin{array}{l}\frac{x}{x+y}+\frac{y}{x-y}\\ =\frac{x\left(x-y\right)}{\left(x+y\right)\left(x-y\right)}+\frac{y\left(x+y\right)}{\left(x-y\right)\left(x+y\right)}\end{array}$

Useaddition rule of fraction.

  $\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}$

To add fractions with the same denominator, add their numerators and write the result over the common denominator.

  $\begin{array}{l}=\frac{x^2-xy+xy+y^2}{\left(x+y\right)\left(x-y\right)}\\ =\frac{x^2+y^2}{\left(x+y\right)\left(x-y\right)}\end{array}$

Hence the simplified form of the expression is $\frac{x^2+y^2}{\left(x+y\right)\left(x-y\right)}$ .