Given the primal LP: minimise $Z=3x_1+5x_2$ subject to $x_1+2x_{2}ge8$ $3x_1+x_{2}le7$ $x_1,x_{2}ge0$. Find the dual problem, solve the dual using the simplex method, and determine the optimal value of the primal.

Step-by-step solution

1. Write primal in standard form with inequalities of "+ and "-": - Constraint 1: $x_1+2x_2-s_1=8$ with $s_{1}ge0$ (since $ge$)\n- Constraint 2: $3x_1+x_2+s_2=7$ with $s_{2}ge0$\nObjective: minimise $Z=3x_1+5x_2$
2. Formulate dual: For each primal variable $x_j$ create dual variable $y_j$; Primal has 2 variables → dual has 2 constraints; Primal has 2 constraints → dual has 2 variables ($y_1$ from first constraint, $y_2$ from second).\nDual form: Maximise $W=8y_1+7y_2$ subject to: $y_1+3y_{2}le3$ $2y_1+y_{2}le5$ $y_1,y_{2}ge0$
3. Set up dual simplex tableau (since constraints are $le$) using standard form: Add slack $s_3,s_4$: $y_1+3y_2+s_3=3$ $2y_1+y_2+s_4=5$
4. Initial tableau: |BV | $y_1$ | $y_2$ | $s_3$ | $s_4$ | RHS | |----|-----|-----|-----|-----|-----| |$s_3$ | 1 | 3 | 1 | 0 | 3 | |$s_4$ | 2 | 1 | 0 | 1 | 5 | |$W$ | -8 | -7 | 0 | 0 | 0 |

Concept tested: Duality and solving dual LP

Published 2 September 2026 · How this solution was prepared and checked

Related questions

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