Question

 Minimize : Z = 3x₁ + 2x₂

Subject to constraints 

5x₁ + x₂ ≥ 10

2x₁ + 2x₂ ≥ 12

x₁ + 4x₂ ≥ 12

x₁ , x₂ ≥ 0

Answer 1

In equation	Equation	Points	Region
5x1 + x2 ≥ 10	5x1 + x2 = 10	A (2,0) , B (0 , 10)	Non-origin
2x1 + 2x2 ≥ 12	2x1 + 2x2 = 12	C (6 , 0) ,  D (0 , 6)	Non-origin
x1 + 4x2 ≥ 12	x1 + 4x2 = 12	E (12 , 0) ,  F (0 , 3)	Non-origin
x1 , x2 ≥ 0			I st quadrant

BQRE - unbounded feasible region 

At B (O, 10), Z = 3 (0) + 2 (10) = 20 

Q (l, 5 }, z = 3 (1) + 2 (5) = 13 

R (4, 2), Z = 3 _(4) + 2 (2) = 16

E (12, 0), Z = 3 (12) + 2 (0) = 36 

Z is minimum at Q (1 , 5) and the minimum value is Z = 13.