Consider the following linear program:

min$-2x_1-3x_2-2x_3-3x_4$

subject $to-2x_1-x_2-3x_3-2x_4-8$

$3x_1+2x_2+2x_3+x_{4}7$

$x_1,x_2,x_3,x_{4}0$

Write the problem $(1)$ in standard form.

Write down the dual of problem $(1).$

Solve the dual of problem $(1)$ graphically.

Use duality to verify that the optimal solution of problem $(1)$ is $x_1=0,x_2=2,$

$x_3=0,x_4=3.$