Problem $3$

Consider the following linear program:

Min $13$x$1+10$x$2+6$x$3$

St$.5$x$1+$ x$2+3$x$3=8$

$3$x$1+$ x$2=3$

x$1>=0$

x$2>=0$

x$3>=0$

a$)$ Formulate the dual linear program

b$)$ The optimal solution of the primal is $(1,0,1).$ Using complementarity slackness,

find the solution to the dual linear program.

c$)$ Do the optimal objective function values of the primal and dual match?