TITLE: Easiest way to solve this system of equations
QUESTION [15 upvotes]: I have these two equations:
$$x=\frac{ab(1+k)}{b+ka}\\ y=\frac{ab(1+k)}{a+kb}$$
where $a,b$ are constants and $k$ is a parameter to be eliminated. 
A relation between $x,y$ is to be found. What is the best way to do it? Cross multiplying and solving is a bit too hectic. Is there a way we can maybe exploit the symmetry of the situation? Thanks!!

REPLY [4 votes]: Alternatively, note that $$\frac xy=\frac{a+kb}{b+ka}\implies (bx-ay)=k(by-ax)\implies k=\frac{bx-ay}{by-ax}$$ and equate with @dxiv's answer.