If I had an underdetermind set of simultanous equations, to take a simple example:

x + 2y = 10
y - z = 2

Is there a way of predicting the number of possible solutions for x,y and z? Or a numerical method for finding a possible solution?

I have a complex linear system with around over 60 variables, and I can produce 27 simulateous equations that are always true for my system This is the ultimate undetermind system!!

Other than doing a trial and error brute force-type search, is there an area of maths that deals with such a problem? Is there a way of defining a search space, and at least predicting where a possible solution would lie?

thank you very much for your help

best regards