Simplify the following:$\frac{x^{-1}+y^{-1}}{x^{-1}}+\frac{x^{-1}-y^{-1}}{x^{-1}}$
Given :
The given expression is $\frac{x^{-1}+y^{-1}}{x^{-1}}+\frac{x^{-1}-y^{-1}}{x^{-1}}$
To do :
We have to find the value of $\frac{x^{-1}+y^{-1}}{x^{-1}}+\frac{x^{-1}-y^{-1}}{x^{-1}}$
Solution :
$\frac{x^{-1}+y^{-1}}{x^{-1}}+\frac{x^{-1}-y^{-1}}{x^{-1}} = \frac{(\frac{1}{x} + \frac{1}{y})}{\frac{1}{x}} + \frac{(\frac{1}{x} - \frac{1}{y})}{\frac{1}{x}} $
$ = \frac{\frac{x+y}{xy}}{\frac{1}{x}} + \frac{\frac{y-x}{xy}}{\frac{1}{x}}$
$= \frac{x+y}{y} + \frac{y-x}{y}$
$= \frac{x+y+y-x}{y}$
$= \frac{2y}{2}$
$= 2$
Therefore, the value of $\frac{x^{-1}+y^{-1}}{x^{-1}}+\frac{x^{-1}-y^{-1}}{x^{-1}}$ is 2.
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