What is the most effective way to solve a system of non-linear equations if we know for sure that they have a unique set of roots?
I think that all mathematicians would say that it is very important to develop a toolbox of problem solving approaches because all problems have unique subtleties.
Being able to look at problems from different angles is very important as there is typically never a one-size fits all across all problems (there are cases where generalizations are possible in some areas).
However, with non-linear equations, all sorts of strange things can happen. For your question, you might approach it in an algorithmic way where you try the standard approaches and see if there is a closed form solution, try graphing, transformations, numerical methods...
There are many wonderful books on the subject and there are programs you can use (both commercial and free).
If you could provide a specific example of the sort of problem you are actually solving, the group could perhaps provide some more specific details regarding an approach.