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

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?

• Line 2, middle term. You changed a + into a - (denominator!) – imranfat Jun 9 '18 at 18:07
• It's a local typo, back to $+$ in the next line. – Arnaud Mortier Jun 9 '18 at 18:08
• @ArnaudMortier I didn't check that, ok! – imranfat Jun 9 '18 at 18:08
• The last expression can be simplified further – imranfat Jun 9 '18 at 18:11
• Multiplying every term by $xy$ as soon as you've flipped the bracketed expressions would lead you to solution quicker. – Džuris Jun 9 '18 at 21:35

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