Solver's default settings greatly limit its accuracy, but I can usually tweak
it to get what I want. Goal Seek seems to permit far less user control.
However, I suspect that most algorithms would perform poorly when asked to
numerically solve an n-dimensional problem that analytically reduces to an
n-k (k0) dimensional problem. Do the analytical reduction as Harlan
suggested, then apply appropriate methods to the problem that remains.

If the algebra is tedious, download Maxima
http://maxima.sourceforge.net
which is a free symbolic math program. In Maxima,
solve ([21*x+ 6*y = 10, (7*x-20)^2+(6*y+10)^2=200], [x,y]);
returns the two solutions to the reduced quadratic equation
[x=10/7,y=-10/3],[x=6/7,y=-4/3]
With more complicated equations, you might get a reduction that actually
requires a numerical solution, but with a clear picture of the actual
dimensionality of the problem, Solver will perform much better.

Jerry

"David A. Heiser" wrote:

Using Solver to solve these equation sets is a disaster. You cannot rely on
the accuracy of the results, which generally are only accurate to 2-3
decimal digits. Most of the time it is not accurate to any decimal digit.
The algorithm is faulty.

David Heiser