Maximize z 7x 4y subject to 3x y lessthanorequalto 14 2x

Maximize: z = 7x + 4y subject to: 3x - y lessthanorequalto 14 2x + y greaterthanorequalto 12 x greaterthanorequalto 4 y lessthanorequalto 8 Use graphical methods to solve. The maximum value is (Type an integer or a simplified fraction.) The maximum occurs at the point (Type an ordered pair. If the maximum occurs at more than one point, type either answer. Type an integer or a simplified fraction.)

Solution

Notice the endpoints of the region are
(4,4)
(4,8)
(5.2 , 1.6)
(22/3 , 8)

The z = 7x +4y to be maximized

With (4,4), we get :
z = 28 + 16 = 44

With (4,8), we get :
z = 28 + 32 = 60

With (5.2,1.6), we get :
z = 7(5.2) + 4(1.6)
z = 42.8

With (22/3,8), we get :
z = 7(22/3) + 4(8)
z = 250/3 or 83.333333333333333

So, clearly :

Maximum value = 250/3 ---> FIRST ANSWER

And it occurs at (22/3 , 8) ---> SECOND ANSWER