 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/ib/paper-3/20d/b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

minimize subject to 11 y_{1} + 16 y_{2} + 29 y_{3}, - 3 y_{2} + 9 y_{3} \geq 4, y_{1} + 2 y_{2} - 2 y_{3} \geq 1, - 11 y_{1} - 7 y_{2} + 10 y_{3} \geq - 9, y_{1}, y_{2}, y_{3} \geq 0.

The negatives of the final slack payoff coefficients give

y^{*} = (1, 2/3, 2/3)

. Its three dual constraints are equalities, and its objective is

11 + 32/3 + 58/3 = 41

. Equality with the primal objective proves optimality by weak duality, independently of tableau sign conventions. It is also the complementary slackness certificate, with all primal variables and all dual variables strictly positive.

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