Solvability of linear equations and its economic consequences

Based upon the solvability theorem of the system of linear equations we will establish the Kuhn-Fourier theorem and the Farkas-Minkowski lemma and related results. Our analyses are elementary and self-contained and perhaps the most basic way to deal with the problem. This will also clarify and simplify the nature of the former analyses based on linear inequalities. Then for applications we give simple proofs of the duality theorem for the linear programming and the minimax theorem in game theory based upon our results.