$4\mathrm{.}6-1.$ Consider the following problem.

Maximize $,Z=2x_1+3x_2,$

subject to

$x_1+2x_{2}4$

$x_1+x_2=3$

and

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

D$,$I $($a$)$ Solve this problem graphically.

$($b$)$ Introduce an artificial variable to reformulate this problem as a convenient artificial problem for preparing to apply the simplex method.

$($c$)$ Describe the enlarged feasible region that has been generated by introducing this artificial variable.

$($d$)$ Explain what needs to happen to the value of the artificial variable in order to guarantee that the optimal solution for the artificial problem will also be the optimal solution for the real problem.