A naive solution would be to optimize a dummy variable via linear programming and see if a result is returned. I imagine there must be a more direct way.
[Math] How to tell if a system of linear inequalities has a solution
linear algebralinear programming
Best Answer
Most linear programming solvers check for feasibility first (usually this takes as much time as the second phase, which is actually finding the optimum, and uses the exact same algorithm). I would advise reading a standard textbook on the subject (Luenberger is good, Schrijver great if you are interested in integer/mixed problems).