@Markiyan Hirnyk I am still evaluating the possible approaches and may or may not settle on one of the solutions you've proposed.  Therefore, my words do correspond to reality.  Your solutions are not ideal; they are two possible options.

@Carl Love 

This is one way to attempt a solution.  However, to generate an explicit numerical solution we would still have to substitute some specific values in for the remaining variables, in which case we could have arbitrarily done this to begin with, and called fsolve.  Also, as Markiyan Hirnyk pointed out, this is unlikely to perform very well in general.

Still, if one is going for a multifaceted approach then this might be one of the directions one might try to get an explicit numerical solution.  (Eliminate as many variables as you have equations, and then assign explicit numerical values to the remaining variables.)  It's possible that this would find a solution in some situations where fsolve would fail; it depends if fsolve already tries this technique as one of its internal algorithms, or if it goes immediately for numerical methods.

@Markiyan Hirnyk 

These are good ideas.  The ideal solution would be equivalent to fsolve(...) or evalf(solve(...)) in the case where the number of equations matched the number of unknowns.  This is because the Maple built-ins solve and fsolve already work very well (when the number of equations match the number of unknowns), and because Maplesoft will continue to improve fsolve and solve.

So these are good ideas, but not the silver bullet.