# How do you simplify (x/y - y/x)/(1/y + 1/x) and can you use a calculator to simplify?

Jul 4, 2018

$\textcolor{red}{\text{The answer is (x-y)}}$

#### Explanation:

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

On further solving this problem we get,

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

Now dividing both the sides we get, $x y$ as cancelled from both the numerator and denominator.

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

Cancelling $x + y$ from both numerator and denominator we get,

Ans. = $\left(x - y\right)$

Jul 4, 2018

Looking at the numerator

$\frac{x}{y} - \frac{y}{x}$ if we put them over a common denominator

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

Now looking at the denominator

$\frac{1}{y} + \frac{1}{x}$ and put this over a common denominator

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

So we have

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

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

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

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

$x - y$