Consider the following LP:

max z $=2$x$1$ x$2+$ x$3$

s$.$t$.$ x$1+$ x$2+$ xProblem $5(20$ points$)$ Consider the following LP:

max,$z=-2x_1-x_2+x_3$

$s.t.x_1+x_2+x_{3}3$

$,x_2+x_{3}2$

$,x_1+x_3=1$

$,x_1,x_2,x_{3}0$

$($a$)$ Find the dual of this LP$.$

$($b$)$ After adding a slack variable $s_1,$ substracting an excess variable $e_2,$ and adding artificial

variable $a_2$ and $a_3,$ row $0$ of the LP$\text{'}$s optimal tableau is found to be

$z+4x_1+e_2+(M-1)a_2+(M+2)a_3=0.$

Find the optimal solution to the dual of this LP$.$

x$2+$ x$3>=2$

x$1+$ x$3=1$

x$1,$ x$2,$ x$3>=0$

$($a$)$ Find the dual of this LP$.$

$($b$)$ After adding a slack variable s$1,$ substracting an excess variable e$2,$ and adding artificial

variable a$2$ and a$3,$ row $0$ of the LP$$s optimal tableau is found to be

z $+4$x$1+$ e$2+($M $1)$a$2+($M $+2)$a$3=0.$

Find the optimal solution to the dual of this LP$.$