Solve the following linear programming problem graphically :

Maximize

Subject to the constraints

To solve the linear programming problems graphically first we need to plot the graphs of the constraints assuming them as equations and then we have to find the feasible region which is the common region of all the constraints, and at the end we have to find the coordinates of the corners of the feasible region and put them one by one in the Z function to check which point makes Z the maximum.

The constraints are,

4x + 6y ≥ 240

6x + 3y 240

x ≥ 10

x ≥ 0 and y ≥ 0

and we have to maximize Z = 7x + 10y

4x + 6y ≥ 240

6x + 3y 240

The coordinates of the corners of the feasible region are now known to us, so let’s put them in the Z function.

Z = 7x + 10y

(10,60) ; Z = 70 + 600 = 760

(30,20) ; Z = 210 + 200 = 410

(10,33.333) ; Z = 70 + 333.333 = 403.333

Therefore the maximum value comes at (10,60), so that is the required point.