Question

Solve the following Linear Programming Problem graphically:

Maximize Z = `(2x)/5 + (3y)/10`

Subject to constraints

2x + y ≤ 1000

x + y ≤ 800

x, y ≥ 0

Answer 1

Line 1 (L₁): 2x + y = 1000

If x = 0, y = 1000 → (0, 1000)

If y = 0, x = 500 → (500, 0)

Line 2 (L₂): x + y = 800

If x = 0, y = 800 → (0, 800)

If y = 0, x = 800 → (800, 0)

Non-negativity: x ≥ 0, y ≥ 0

To find where L₁ and L₂ intersect, we subtract the equations:

(2x + y) − (x + y) = 1000 − 800

x = 200

Substituting x = 200 into L₂:

x + y = 800

200 + y = 800

y = 600

The intersection point is (200, 600).

Corner Point (x, y)	Z = 0.4x + 0.3y	Value of Z
(0, 0)	0.4(0) + 0.3(0)	0
(500, 0)	0.4(500) + 0.3(0)	200
(200, 600)	0.4(200) + 0.3(600)	260 (Max)
(0, 800)	0.4(0) + 0.3(800)	240