Consider the following linear programming $($LP$)$ problem.

Maximize $z=-5x_1+5x_2+13x_3$

subject to:

$-1x_1+1x_2+3x_{3}20$

$12x_1+4x_2+10x_{3}90$

$x_{1}0,x_{2}0,x_{3}0.$

a$.$ Construct the augmented form of this LP by introducing slack variables. $(10$ pts$)$

b$.$ The final simplex tableau for the above LP is given below.

$\backslash$table$[[\backslash$table$[[$Basic$],[$Variable$]],$Eq$,$Coefficient of:$,\backslash$table$[[$Right$],[$Side$]]],[$z$,$x$\_(1),$x$\_(2),$x$\_(3),$x$\_(4),$x$\_(5)],[$Z$,(0),1,0,0,2,5,0,100],[$x$\_(2),(1),0,-1,1,3,1,0,20],[$x$\_(5),(2),0,16,0,-2,-4,1,10]]$

Now suppose that the right$-$hand side of the second constraint is changed from $b_2=90$ to $b_2=70.$ Using the sensitivity analysis procedure covered in class, explain why the current solution is not optimal. Construct the revised tableau and provide the next step to follow $($should primal simplex or dual simplex method be used to find the optimal solution?$)(20$ pts$)$

c$.$ Following the method you identified in part b$),$ find the optimal solution and the optimal value for the revised problem. Provide also the revised final tableau. $(30pts).$ DO NOT SOLVE WITH COMPUTER SOLVING.