I want to solve the following two equations in two unknowns:
The unknowns are $x$ and $y$. Please help me.
The second equation can be solved for $y$: $$y=x/(1+(b/x))=x^2/(x+b)\tag1$$ Now rewrite the first equation as $$-2a-bcy^2+2cx^2(x-y)=0$$ and use (1) to get an equation involving only $x$. Come back when you've done that, and we'll see what we can do from there.
EDIT: Maple says $x=\alpha$, $y=\alpha^2/(b+\alpha)$, where $\alpha$ is a root of $$bcz^4+2b^2cz^3-2az^2-4abz-2ab^2=0$$