Hey Fratricide.

The idea is very similar to the pair-wise case with one twist.

What you do is you do all pair-wise solutions and find all solutions that are common to all of them. If you don't find any common solution to all pair-wise solutions then it means that the system of equations is in-consistent just like the case when you have an inconsistent system in linear algebra.

Doing this efficiently without a computer is tough, but I would look at the linear and quadratic systems differently and then together.

Linear algebra allows us a way to do this in a generalized form but since you have quadratics, you will need to do this by first principles.

Mathematically this amounts to taking the intersection of all solutions given by all the pair-wise solutions (i.e. the solution to (x,y) where x != y and x,y = 1 to 5 in your case) and using this result as the solution of this system of equations.