Consider the following linear programming problem

$,$minw$=45x+30y$

subject $tox+4y12$

$,2x+2y12$

$,5x+2y20$

$,7x+y14$

$,x,y0.$

$($a$)$ Use the graphical method to solve the problem. In doing so you must clearly indicate the feasible region and the optimal vertex, making good use of objective lines.

$($b$)$ Discuss what happens to the optimal solution as the absolute value of the gradient of the objective function decreases?