I have the following set of equations/inequalities: $$ \vec{a}^T \vec{x} + b = 0 \\ M \vec{x} + \vec{c} \le 0 $$ I am trying to find whether a solution exists or not. I do not care about uniqueness or the actual solution. I am aware that this can be done by using optimization algorithms, such as simplex. However, I am looking for another way, e.g. by computing the rank.
I also found out about https://en.wikipedia.org/wiki/Constrained_generalized_inverse. I thought about introducing slack variables to create a single system of linear equations, but the subspace of positive slack variables is not complete.