Solution

• From 2x - 3y <= 5, if we put x = 0, then y = -(5/3) and if we put y = 0, then x = 5/2. Hence, we get (0, -1.67) and (2.5, 0).
  Note - (We will reject (0, -1.67) as here y < 0).
• From x + 3y <= 11, if we put x = 0, then y = 11/3 and if we put y = 0, then x = 11. Hence, we get (0, 3.67) and (11, 0).
  Note - (Point (11, 0) will be rejected because it is outside the shaded region in the below graph).
• From 4x + y <= 15, if we put x = 0, then y = 15 and if we put y = 0, then x = 15/4. Hence, we get (0, 15) and (3.75, 0).
• Now, we will plot the graph using these points.
  
   
• Our Maximum value lies in the shaded region ABCDE.
• A, B and C are corner points whereas D and E are intersection points. We need to find the intersection points.
• To find point D, we will use 4x + y <= 15 and x + 3y <= 11. We will multiply x + 3y <= 11 by 4 and subtract it from 4x + y <= 15. We will get y = 29/11. We will put this y in 4x + y <= 15 and get x = 34/11. Hence, point D = (34/11, 29/11) = (3.09, 2.63).
• Similarly, for point E, we need 4x + y <= 15 and 2x - 3y <= 5. We will get x = 25/7 and y = 5/7. Hence, point E = (25/7, 5/7) = (3.57, 0.71)
• Now, we will put all the values of point A, B, C, D and E in z = 6x + 5y and get the value of z:
  • A(0, 3.67) = 18.35
  • B(0, 0) = 0
  • C(2.5, 0) = 15
  • D(3.09, 2.63) = 31.69
  • E(3.57, 0.71) = 24.97
• We can see that point D(3.09, 2.63) has the maximum value. Hence, 31.69 ~ 31.72 is the answer.