[newbie] algsys: tried and failed to reduce system to a polynomial in one variable; give up.

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> you are trying to solve analytically 18th degree polynomials with 3 variables. I think your only choice will be to use numerical methods.

When I looked at your equations I saw nested square roots of polynomials. Generally if equations are so complex that there is no
clear path to a hand solution, you shouldn't expect a CAS to solve the problem. And if there is a huge closed form solution, likely it is
amusing but not useful--have you ever seen Physical Review  print a single expression that spans several pages? In addition
to a numerical method suggested by Jaime, maybe you can identify some terms that are small--Maxima's multivariable Taylor function
might help in such cases. You will see plenty of such approximations printed in the literature.

One of my favorite numerical analysis problems has something to do with a parabolic orbit in an inverse square law force. Determining
something (can't recall exactly what) requires the solution of a fairly tame 3rd degree polynomial. Oops, expressing the solution
in radicals gives rise to an expression that can be ill conditioned for numerical evaluation.  Even the quadratic formula has this same
ill conditioning for some inputs--for the quadratic case there well known rearrangement that renders it well conditioned--maybe the same
is true for the cubic case, but the truth is that a few iterations of the Newton method is fast, accurate, and far less error prone than a
cute (but crabbed) rearrangement of the solution of cubic equation in radicals.

--Barton