Question

Minimize = 20x + 10y

subjected to x + 2y ≤ 40, 3x + y ≥ 30, 4x + 3y ≥ 60

x ≥ 0, y ≥ 0

Answer 1

First we draw the lines

x + 2y = 40, 3x + y = 30, 4x + 3y = 60

The feasible region(shaded region) is the bounded region "EAQPA''

The vertices of the region are 

E(15,0), A(40,0), Q(4,18) and P(6,12) 

Given objective function is 

z = 20x + 10y

At, E(15,0), z = 300

At, A(40,0), z = 800

At, Q(4,18), z = 260

At, P(6,12), z = 240

Hence, Z_min = 240 at x = 6, y = 12