I have a lot of systems of equations and inequalities of the following form:

$$ a_{1,1}x+a_{1,2}y+a_{1,3}z+a_{1,4}w = 2 $$ $$ \ldots $$ $$ 0 < x < 2 $$ $$ 0 < y < 2 $$ $$ 0 < z < 2 $$ $$ 0 < w < 2 $$

There are always at least two equations, and I probably won't consider cases with more than twenty equations. All coefficients $a_{i,j}$ are positive integers and some can be zero. We also have the property that $\sum_{j=1}^4a_{i,j}\geq3$ for all $i$. The solutions are real numbers.

I don't need to solve these systems, but I need to be able to tell whether there exists a solutions. If it isn't possible to tell for each system whether it is consistent or not, any method which identifies as many inconsistent systems as possible is greatly appreciated.

I have a few hundred millions of these systems, so I'm specifically looking for things that can easily be turned into a program. (I know the basic techniques to do this by hand, and am looking for some handy tricks that can be done by a computer. I have some programming experience, but not really with programming this kind of problems.)