Suppose I want to prove that a variable posesses a certain property (e.g. is a square). For example if I wanted to prove that $x$ in $\frac{x^2+y^2+1}{xy} = k$ has the property of being a square (It is obviously false, but suppose it is true ). Is it then necessary to fix $x$ and look at the pairs $(y,k)$ (which satisfy the equation), or look at the pairs $(x,y)$ and fix k? The latter is the only version I have seen so far.
( source to the vieta jumping method : www.yimin-ge.com/doc/VietaJumping.pdf )