Is there an adjective to describe systems of equations which is neither underdetermined nor overdetermined? What might I call a system of equations in which the number of equations equals the number of free variables? In other words, if a system of equations is neither underdetermined nor overdetermined, then what is it?
 A: There is no standard terminology for this specific situation, although "critical" or "critically determined" as others have suggested can be appropriate.  But the comment thread on the question should be a good indication that it's best to avoid using a term like this (i.e., applying a word like "critical" in a context where it sort of makes sense but still isn't standard) without previously defining it.  I say this because two different people proposed two different, albeit very similar, answers within two minutes of each other.
I think the best course of action would be to define the term the first time you use it.  For example, if you're writing a scholarly article then you can say, "...for this critically determined system, i.e., neither underdetermined nor overdetermined..." or something to that effect.
A: A system of equations in which the number of equations equals the number of free variables is often referred to as the critical case. It is the case between overdetermined and underdetermined, where for each variable giving a degree of freedom, there is a corresponding constraint removing a degree of freedom.
