Consider the following LP:

maxz$=x_1+2x_2$

$s.t.3x_1-x_{2}5$

$2x_1+x_{2}6$

$x_{1}0,x_2$ urs

$($a$)(3$ points$)$ Find the dual of the LP$.$

$($b$)(5$ points$)$ At optimality, we know that $x_1^{**}>0$ and $x_2^{**}>0.$ Use complementary slackness to find the optimal dual solution.

$($c$)(2$ points$)$ Based on the answer in part $($b$),$ how much would you be willing to pay for an additional unit of resource in the second constraint?

Hint: Assume that the primal LP is a maximization problem and the dual LP is a minimization problem. Then we have the following rules for obtaining the dual:

$\backslash$table$[[$Primal$,$Dual$],[$Equality constraint,Unrestricted$-$sign variable$],[$Inequality constraint $)(,$Nonnegative variable$],[$Unrestricted$-$sign variable,Equality constraint$],[$Nonnegative variable,Inequality constraint $)($