Where did I make a mistake in my simplification of the algebraic expression?

Question

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 $$2x$$

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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?

Answer 1

There is no mistake, simply note that $$\frac{2(1+xy^{-1})}{x^{-1}+y^{-1}}=\frac{2x(x^{-1}+y^{-1})}{x^{-1}+y^{-1}}=2x$$

Answer 2

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.