Given a system that contains a mixture of linear and nonlinear equations, under what conditions can we guarantee that a solution will exist and that it will be unique?
For example, the system \begin{align} a^2+b^2&=c^2 && (1) \\ a+b+c&=1000 && (2) \end{align} has a unique solution $$(200, 375, 425)$$ if we additionally restrict $a$, $b$, and $c$ to be integers (some of you may recognize this as problem 9 from Project Euler).
However, if we change $(2)$ to $$a+b+c=k$$ for some arbitrary integer $k$, for what choices of $k$ will we achieve a unique solution? More generally, when we have ANY system where some equations are linear and others are nonlinear, are there circumstances that allow us guarantee similar outcomes?
