If a system of equations has no solutions, it implies a partial solution satisfying one of the equations does not satisfy at least one of the other equations. And this happens for all such single solutions, so, as a result, the set of solutions satisfying the system is empty - there are no solutions. That's why the term "inconsistent" is used. The opposite case happens when you find a solution for one equation and test it and it works (i.e., it is a solution) for the second equation as well (and all others) - then the term is the solution is "consistent."
