Duality theory establishes that every linear programming problem (the primal) has an associated dual problem. The optimal value of the primal equals the optimal value of the dual (strong duality theorem). If the primal is: maximize c^T x subject to Ax ≤ b, x ≥ 0, then the dual is: minimize b^T y subject to A^T y ≥ c, y ≥ 0. Duality provides economic interpretations (shadow prices), bounds on optimal values, and alternative solution methods. Complementary slackness conditions link primal and dual optimal solutions.