Are there any strategies for solving Diophantine equations where the solutions can be any rational number, not just an integer, besides substituting $x=p/q$ and $y=r/s$, with $p,q,r,s$ integers with $\gcd(p,q)=\gcd(r,s)=1$?
How can one, in general, use the condition $\gcd(p,q)=\gcd(r,s)=1$ well?
I'm mainly interested in the type that appear on contests, so please don't suggest to use modular or elliptic curves to solve it, unless it is a special form that doesn't need the massive calculations it usually needs.