In the world of Diophantine analysis, there are single equation problems (e.g., “Find all solutions $(x,y)$ such that $x^3=y^2+1$”), double equation problems (e.g., “Find all numbers $p$ such that $p+1$ and $p^2+1$ are both the double of a square”), and problems involving three or more equations which share one or more of the variables.
What, if any, are the principal methods of [non-trivially] “upgrading” one type to a “higher” type? Can every single equation problem be [non-trivally] turned into a multiple-equation problem?
Note: By “non-trivially”, I mean not just rational transformations like writing an odd number $x$ as $2u+1$. Rather, I mean taking a single equation which [apparently] provides 'exactly $N$' units of information about the numbers/variables involved and turning it into two or more equations where the total amount of information is greater than '$N$'.