Problem $5(20$ points$)$ Consider the following $LP$ :

maxz$=-2x_1-x_2+x_3$

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

$x_2+x_{3}2$

$x_1+x_3=1$

$x_1,x_2,x_{3}0$

$($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$.$