????????

Hello, isabelle2hot!

\(\displaystyle \L\frac{x^{-1}\,-\,y^{-1}}{x^{-1}\,+\,2y^{-1}}\;=\;\)(x + 2y)/(y - x) ?? . . . . they don't "move" like that!
We have: .\(\displaystyle \L\frac{\frac{1}{x}\,-\,\frac{1}{y}}{\frac{1}{x}\,+\,\frac{2}{y}}\)

Multiply top and bottom by \(\displaystyle xy:\;\;\L\frac{xy\left(\frac{1}{x}\,-\,\frac{1}{y}\right)}{xy\left(\frac{1}{x}\,+\,\frac{2}{y}\right)} \;=\;\frac{xy\left(\frac{1}{x}\right)\,-\,xy\left(\frac{1}{y}\right)}{xy\left(\frac{1}{x}\right)\,+\,xy\left(\frac{2}{y}\right)}\)

. . which simplifies to: .\(\displaystyle \L\frac{y\,-\,x}{y\,+\,2x}\)
 
If you are unsure if the statement is true you can always check by subing in number for x & y, for simplicity lets choose x=0 and y =0. the statement is not true so it is not right,

OOPS, i took that as x-1 not x^-1
 
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