Consider the linear programming problem.

max,$Z=2x_1+3x_2+2x_3+x_4+4x_5$;$,$

subject $to{x}_1+4x_2+3x_3+12x_4+3x_5,40,$

$,x_1+5x_2+8x_3+4x_4+x_5,30,$

$,x_{1}0,x_{2}0,x_{3}0,x_{4}0,x_{5}0.$

$($a$)$ State the dual of the above primal problem.

$($b$)$ Sketch the feasible region for the dual program and solve the dual problem graphically.

$($c$)$ Write down one of the useful properties between primal and dual problems.

$($d$)$ Hence solve the primal problem.