$7-$ Answer following questions by considering given LP and its optimal tableau. $(a_2$ and $a_3$ are the

artificial variables for constraints $2$ and $3,s_1$ is the slack variable for constraint $1,$ and $e_2$ is the surplus

$($excess$)$ variable for constraint $2)$

a$.$ Find the range of values of the objective function coefficient for $x_2$ for which the current basis

remains optimal.

b$.$ Find the range of values of $b_3$ for which the current basis remains optimal.

c$.$ Find the new optimal solution if $b_2=35.$

Max $z=3x_1+4x_2+x_3$

$s.t.$

$x_1+x_2+x_{3}50$

$2x_1-x_2+x_{3}15$

$x_1+x_2=10$

$x_1,x_2,x_{3}0$