A linear programming model is given as follows:

Maximize $Z=6\mathrm{.}5x_1+10x_2$

Subject $to2x_1+4x_{2}40$

$x_1+x_{2}12$

$x_{1}8$

$x_1,x_{2}0$

$($Accept both geometric method and Excel Solver$)$

$($a$)$ Define the feasible solution area and obtain the optimal solution graphically.

$($b$)$ Write the model in the standard form using slack or surplus variables for all the constraints.

$($c$)$ Compute the values of the slack$/$surplus variables at the solution in $($a$).$

$($d$)$ If the third constraint, $x_{1}8,$ is changed to $x_{1}8,$ what effect will be$?$