Question

The optimal value of the linear programming problem
Maximise $z = 2x + 3y$
subject to
$5x + 4y ≤ 20,$
$ 3𝑥 + 5𝑦 ≤ 15,$
$ 2𝑥 + 𝑦 ≤ 4,$
$ 𝑥, 𝑦 ≥ 0,$
is

• A. 4
• B. 64/ 7
• C. 9
• D. 72/ 7

Answer 1

The Correct Option is B

Solving a Linear Programming Problem 

Objective Function:

Maximize Z = 2x + 3y

Subject to Constraints:

• 5x + 4y ≤ 20
• 3x + 5y ≤ 15
• 2x + y ≤ 4
• x, y ≥ 0

Step 1: Graphing the Constraints

We rewrite the constraints as equalities to determine their boundary lines:

• 5x + 4y = 20 → Intercepts: (4,0) and (0,5)
• 3x + 5y = 15 → Intercepts: (5,0) and (0,3)
• 2x + y = 4 → Intercepts: (2,0) and (0,4)

These constraints, along with x ≥ 0 and y ≥ 0, define the feasible region.

Step 2: Finding the Intersection Points (Vertices)

We determine the vertices by solving the equations pairwise.

1. Intersection of 5x + 4y = 20 and 3x + 5y = 15

Solving:

• Multiply first equation by 5: 25x + 20y = 100
• Multiply second equation by 4: 12x + 20y = 60
• Subtract: 13x = 40 → x = 40/13
• Substituting in 5x + 4y = 20: y = 15/13

Vertex: (40/13, 15/13)

2. Intersection of 5x + 4y = 20 and 2x + y = 4

Solving:

• Express y: y = 4 - 2x
• Substituting in 5x + 4y = 20: x = 4/3, y = 4/3

Vertex: (4/3, 4/3)

3. Intersection of 3x + 5y = 15 and 2x + y = 4

Solving:

• Express y: y = 4 - 2x
• Substituting in 3x + 5y = 15: x = 5/7, y = 18/7

Vertex: (5/7, 18/7)

Step 3: Evaluating the Objective Function at Each Vertex

• At (40/13, 15/13): 
  Z = (2 × 40/13) + (3 × 15/13) = 125/13
• At (4/3, 4/3): 
  Z = (2 × 4/3) + (3 × 4/3) = 20/3
• At (5/7, 18/7): 
  Z = (2 × 5/7) + (3 × 18/7) = 64/7

Step 4: Conclusion

The maximum value of Z = 64/7 occurs at the vertex (5/7, 18/7).