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I need to simplify $$\left(\frac{x^{-1}+y^{-1}}{yx^{-1}+xy^{-1}}\right)^{-1}+\left(\frac{x^{-1}+y^{-1}}{2}\right)^{-1}-\frac{x^{-1}-y^{-1}}{x^{-1}y^{-1}}$$ The conditions are $$xy\neq0$$ $$x\neq-y$$ And the solution is $$2x$$


My attempt $$\left(\frac{x^{-1}+y^{-1}}{yx^{-1}+xy^{-1}}\right)^{-1}+\left(\frac{x^{-1}+y^{-1}}{2}\right)^{-1}-\frac{x^{-1}-y^{-1}}{x^{-1}y^{-1}}$$ $$\frac{yx^{-1}+xy^{-1}}{x^{-1}+y^{-1}}+\frac{2}{x^{-1}+y^{-1}}-\frac{x^{-1}-y^{-1}}{x^{-1}y^{-1}}$$ $$\frac{2+yx^{-1}+xy^{-1}}{x^{-1}+y^{-1}}-((x^{-1}-y^{-1})(xy))$$ $$\frac{2+yx^{-1}+xy^{-1}}{x^{-1}+y^{-1}}-(y-x)$$ $$\frac{2+yx^{-1}+xy^{-1}}{x^{-1}+y^{-1}}-\frac{(y-x)(x^{-1}+y^{-1})}{x^{-1}+y^{-1}}$$ $$\frac{2+yx^{-1}+xy^{-1}}{x^{-1}+y^{-1}}-\frac{yx^{-1}-xy^{-1}}{x^{-1}+y^{-1}}$$ $$\frac{2(1+xy^{-1})}{x^{-1}+y^{-1}}$$ Where's my mistake?

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    $\begingroup$ Line 2, middle term. You changed a + into a - (denominator!) $\endgroup$ – imranfat Jun 9 '18 at 18:07
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    $\begingroup$ It's a local typo, back to $+$ in the next line. $\endgroup$ – Arnaud Mortier Jun 9 '18 at 18:08
  • $\begingroup$ @ArnaudMortier I didn't check that, ok! $\endgroup$ – imranfat Jun 9 '18 at 18:08
  • $\begingroup$ The last expression can be simplified further $\endgroup$ – imranfat Jun 9 '18 at 18:11
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    $\begingroup$ Multiplying every term by $xy$ as soon as you've flipped the bracketed expressions would lead you to solution quicker. $\endgroup$ – Džuris Jun 9 '18 at 21:35
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There is no mistake, simply note that $$\frac{2(1+xy^{-1})}{x^{-1}+y^{-1}}=\frac{2x(x^{-1}+y^{-1})}{x^{-1}+y^{-1}}=2x$$

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I don't believe you have made a mistake. Multiply the numerator and denominator by $xy$ and you should be able to see how it simplifies from there.

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