Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 3 / 21H / b / Solution

Introduce slack variables $s_1,s_2,s_3\ge 0$ after rewriting the third inequality as $x_1-x_2+x_3\le 2$. The initial dictionary isIn the simplex method, $x_3$ enters first because it has the largest positive objective coefficient. The ratio test makes $s_3$ leave, and substitution givesNext $x_2$ enters and $s_2$ leaves. Solving for $x_2$ produces the final objective rowAll reduced costs are now nonpositive, so the dictionary is optimal at the nonbasic values $x_1=s_2=s_3=0$. ThereforeThe first constraint has slack $s_1=56/5$, while the second and third constraints are active.