Question

Solve the following linear programming problem graphically:

Minimize Z = 13x − 15y

Subject to constraints:

x + y ≤ 7,

2x − 3y + 6 ≥ 0,

x ≥ 0, y ≥ 0

Answer 1

• x + y ≤ 7,
• 2x − 3y + 6 ≥ 0, ⇒ y ≤ `(2x + 6)/3`
• x ≥ 0, y ≥ 0

The corner points of the shaded feasible region are:

1. Origin: (0, 0)
2. x-intercept of x + y = 7: (7, 0)
3. Intersection of x + y = 7 and 2x − 3y = −6:
   1. From the first: x = 7 − y
   2. Substitute:
      2(7 − y) − 3y = −6
      14 − 5y = −6
      5y = 20
      y = 4, x = 3
4. y-intercept of 2x − 3y = −6: (0, 2)

Evaluate Z = 13x − 15y:

1. Z (0, 0) = 0
2. Z (0, 2) = 13(0) − 15(2)
   = −30
3. Z (3, 4) = 39 − 60
   = −21
4. Z (7, 0) = 91

So, the minimum value is −30 at (x, y) = (0, 2).