Find the real numbers x and y if $\displaystyle \frac{1}{x+iy}+\frac{1}{1+2i}=1 $

I started like this :

Multiply their respective conjugates for both terms :

i got $\displaystyle \frac{x-iy}{x^2+y^2}+\frac{1-2i}{5}=1$

continue solving from here ,

$\displaystyle

\frac{x^2+y^2+5x}{5x^2+5y^2}-[\frac{5y+2x^2+2y^2}{5x^2+5y^2}]i=1

$

so i ot 2 equations here :

$\displaystyle 5x=4x^2+4y^2 $

$\displaystyle 5y+2x^2+2y^2=0$

I am not sure how to continue from here .. not even sure if my above working is correct . THanks .