TITLE: Where did I make a mistake in my simplification of the algebraic expression?
QUESTION [9 upvotes]: 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?

REPLY [3 votes]: 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.