Question

Minimize Z = 24x + 40y subject to constraints

6x + 8y ≥ 96, 7x + 12y ≥ 168, x ≥ 0, y ≥ 0

Answer 1

To draw the feasible region, construct table as follows:

Inequality	6x + 8y ≥ 96	7x + 12y ≥ 168
Corresponding equation (of line)	6x + 8y = 96	7x + 12y = 168
Intersection of line with X-axis	(16, 0)	(24, 0)
Intersection of line with Y-axis	(0, 12)	(0, 14)
Region	Non-origin side	Non-origin side

Shaded portion XABY is the feasible region, whose vertices are A(24, 0) and B(0, 14).

Here, the objective function is

Z = 24x + 40y

∴ Z at A(24, 0) = 24(24) + 40(0) = 576

Z at B(0, 14) = 24(0) + 40(14) = 560

∴ Z has minimum value 560 at x = 0 and y = 14.