An equilibrium system (also known as a KKT system, a saddle- point system, or a sparse tableau) is a square linear system with a certain structure. G. Strang has observed that equilibrium systems arise in optimization, finite elements, structural analysis, and electrical networks. Recently, G.W. Stewart established a norm bound for a type of equilibrium system in the case that the "stiff- ness" portion of the system is very ill-conditioned. In this paper, we investigate the algorithmic implications of Stewart's result. We show that all standard textbook algorithms for equilibrium systems are unstable. Then we show that a certain hybrid method has the right stability property.