Detecting infeasibility of System of Linear Inequalities Infeasibility of a system of linear inequalities can be detected by using artificial variables and then using an algorithm like the simplex algorithm (or ellipsoid/interior point methods) to find a basic feasible solution. If one cannot be found then the problem is infeasible.

Are there any other methods to detect infeasibility of a the system?
(1) Farkas' Lemma should show the equivalence of the infeasibility problem and LP in strongly polynomial time.

(2) Solving a linear programming problem (as a computational problem) usually means "detect the problem is either infeasible, unbounded, or having an optimal solution, and if the problem has an optimal solution, find one".  In this sense, they are equivalent.