Question

The optimal solution of the L.P.P. \( Z = 8x + 3y \) subject to the constraints \( x + y \leq 3 \), \( 4x + y \leq 6 \), \( x \geq 0 \), \( y \geq 0 \) is

• A. \( x = 0, y = 3 \)
• B. \( x = 0, y = 0 \)
• C. \( x = \frac{3}{2}, y = 0 \)
• D. \( x = 1, y = 2 \)

Hint:

In linear programming problems, always plot the feasible region and evaluate the objective function at the corner points to find the optimal solution.

Answer 1

The Correct Option is D

Step 1: Plot the constraints and find the feasible region.
The constraints are: - \( x + y \leq 3 \) - \( 4x + y \leq 6 \) - \( x \geq 0 \) - \( y \geq 0 \) The feasible region is the region where all the inequalities are satisfied.
Step 2: Calculate the corner points.
By solving the system of equations formed by the constraint lines, we find the corner points of the feasible region.
Step 3: Evaluate the objective function at the corner points.
Substitute the corner points into \( Z = 8x + 3y \) to find the optimal value of \( Z \). The point \( (1, 2) \) gives the maximum value of \( Z \).
Step 4: Conclusion.
The optimal solution is \( x = 1, y = 2 \).