The method of alternating projections was first introduced by Von Neumann in 1933 for finding the projection of a given point onto the intersection of closed subspaces of a Hilbert space. Since its introduction this method has received considerable attention and has found application in many areas of mathematics and physics as well as in other fields of science and engineering. In this paper we show that recent results of the method of alternating projections for general convex feasibility problems may be used to construct a simple algorithm for solving linear programs. This new algorithm is simple to implement, computationally stable and is inherently parallel.