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9.5 (a) We use u as a dual variable for the first constraint and v as a dual variable for the second constraint of the primal LP. The dual is min s.t. 7u + 3v u+v ≥5 u−v ≥1 u ≥ 0, v ≥ 0 (b) The graph of the primal and dual feasible regions are follows: Dual v 5 Primal y 7 6 4 5 3 4 2 3 1 2 0 u 1 2 3 4 5 6 -1 1 x 0 1 2 3 4 5 -1 -2 -3 Using the graphical method, we find that the optimal solution to the primal is (x, y) = (5, 2) and the primal optimal value is 27. Similarly, we find that the optimal solution to the dual is (u, v) = (3, 2), and the dual optimal value is 27. Note that the optimal values of the primal and dual are equal, as the Strong Duality Theorem would imply. 1