Simplify
\(E=\dfrac{\frac{x}{y}-\frac{y}{x}}{\frac{1}{y}+\frac{1}{x}}\)
Conditions for the existance of \(E:\)
\(x
eq 0~~\text{ and}~~y
eq 0~~\text{ and }~~x
eq -y.\)
(otherwise, the one of the denominators would be zero.)
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Simplifying:
\(E=\dfrac{\frac{x}{y}-\frac{y}{x}}{\frac{1}{y}+\frac{1}{x}}\)
Multiply both the numerator and the denominator by \(xy:\)
\(E=\dfrac{xy\cdot \left(\frac{x}{y}-\frac{y}{x} \right )}{xy\cdot \left(\frac{1}{y}+\frac{1}{x} \right )}\\\\\\ E=\dfrac{\frac{xy\cdot x}{y}-\frac{xy\cdot y}{x}}{\frac{xy}{y}+\frac{xy}{x}}\\\\\\ E=\dfrac{x^{2}-y^{2}}{x+y}\\\\\\ E=\dfrac{(x+y)(x-y)}{x+y}\)
Simplify common factors:
\(E=x-y\\\\\\ \therefore~~\boxed{\begin{array}{c}\dfrac{\frac{x}{y}-\frac{y}{x}}{\frac{1}{y}+\frac{1}{x}}=x-y \end{array}}\)
