Question $3(30$ marks$)$

Consider the following LP:

max$-4x_1-2x_2+x_3$

$s.t.$: $x_1+2x_{2}6$

$x_1-x_2+2x_3=8$

$x_1,x_{2}0,x_3$inR

a$)$ Write the dual LP$.$

b$)$ At the optimal solution of the above LP$,$ the slack variable of the $1^{st}$ constraint is equal to $6.$

Use this information to find the value of the dual variable of the constraint.

c$)$ If both constraints are expressed in the form $,$ what conclusion can you draw about the above

LP$?$ Does it have an optimal solution, is it infeasible, or unbounded? Based to your answer,

explain the conclusion that can you draw about the $($new$)$ dual?