Prove using field axioms that $(1/x)+(1/y) = (y+x)/(x*y)$ How can I prove  $(1/x)+(1/y) = (y+x)/(x*y)$ with field axioms? 
I tried to write it like 
$$x{^-}^1+ y{^-}^1
$$
$$x*x{^-}^1+y*y{^-}^1 $$
I am not sure how to process or if this is the right direction.. 

$$\left(x^{-1}+y^{-1}\right)xy=x^{-1}xy+y^{-1}xy=y+y^{-1}yx=y+x.$$
Thus, $$x^{-1}+y^{-1}=\left(x^{-1}+y^{-1}\right)xy(xy)^{-1}=(y+x)(xy)^{-1}.$$
The first line it's $$\left(x^{-1}+y^{-1}\right)(xy)=x^{-1}(xy)+y^{-1}(xy)=(x^{-1}x)y+y^{-1}(yx)=$$
$$=1y+(y^{-1}y)x=y+1x=y+x.$$
Try to write the second part of the proof by yourself. 
