Probably a simple question, but I can't find an answer anywhere, not even in the suggested questions with similar titles. It might also be that I just don't get the correct terminology. This is not really a field with which I'm familiar.
I have a system of equations of the form $aw+bx+cy+dz=e$ and I want to now whether there is a solution. More specific I have this situation:
$Ax=b$
where $A$ is a matrix of $n\times 4$, $x$ is of $4 \times 1$ and $b$ is of $n\times 1$ with $n>4$. $A$ consists solely of non-negative integers, and for my case $b=\begin{bmatrix} 2 \\ \vdots \\2 \end{bmatrix}$.
The question is how I can efficiently implement an algorithm to check whether these equations are consistent, i.e. whether the solution space is non-empty. I'm satisfied with an approximate solution but it may never say that there are no solutions when there are. The other way around is allowed, but I want to exclude as many inconsistent systems as possible.