Help solving the system below

Question

I am trying to solve the system of diophantine equations below where $x,y$ are coprime

\begin{eqnarray} xr^2+ys^2&=&uva\\ xr+ys&= &uvb\\ x+y&=&uc \end{eqnarray}

I solve 2 out of 3 then replace in the remaining one. However, things get messy at this point. I am unable to close the argument.Any hints?

Answer 1

The system may not have any solution. For example, if $(r,s,u,v)=(1,1,5,6)$, then there is no solution for $x,y$. In general we can use that $u$ divides $x+y$, $xr+ys$ and $xr^2+ys^2$. Similarly $v$ divides $xr+ys$ and $xr^2+ys^2$.