Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 35 / 2 / b

Add the slack variable $s_3$ for the forgotten constraint. In the optimal simplex tableau from part (a), $x_1+x_2=4-(s_1+s_2)/3$. Hence the new row in canonical form isAppending it givesThe objective row remains dual feasible, but $s_3=-1$ makes the basic solution infeasible. Apply the dual simplex algorithm: $s_3$ leaves, and the eligible columns are $s_1,s_2$, whose negative coefficients can repair this row. The dual ratio test comparesThus $s_2$ enters. Pivoting on $-4/3$ and eliminating its column givesAll basic values are nonnegative and the objective equation is $z=27/2-s_1-s_3/2$. ThereforeBoth binding constraints have zero slack, and the final objective row proves global optimality. The two nonbasic penalties are strictly positive, so this optimum is unique.