text: sets of non-linear equations in n variables? say you have a set of k equations in n variables.  it is very well known that if the equations are linear, then you are guaranteed a solution so long as you have k=n equations (and they are linearly independent) http://mathworld.wolfram.com/linearsystemofequations.html  now: things get much much weaker if the equations are not linear (polynomial, exponential, trigonometric, hyperbolic etc.). what if anything can you say? how many (extra) equations do you need? (let's completely forget about general existence and uniqueness of solutions. let's only talk about what you need to be guaranteed the chance of at least one solution).  so let's illustrate with an example in x,y,z (could be negative or positive): e^x - (y^2)z = 10 2(x-y)^5 + z^(1.2) = 2 x + (x^2)y + z^3 = 10  don't cheat by using iterative methods. can you say *anything at all* about the solution or solution space?  may be of use: http://scpd.stanford.edu/scpd/courses/academic/crsedesc.asp?crseid=74&sdid=5 well, since the system's first derivative is continuous all we need to look at is the jacobian matrix.  so long as the jacobian is not singular (not equal 0), you can invert the function which implies that a solution exists.