Question

Solve the following linear programming problem graphically: Maximise \( Z = 2x + 3y \), subject to the constraints: \( x + y \leq 6, \, x \geq 2, \, y \geq 3, \, x \geq 0, \, y \geq 0. \)

Hint:

In linear programming, always evaluate the objective function at the corner points of the feasible region.

Answer 1

Step 1: Graph the constraints \( x+y \leq 6, \, x \geq 2, \, y \geq 3, \, x \geq 0, \, y \geq 0 \) on the Cartesian plane. The feasible region is the shaded area defined by these inequalities.
Step 2: Identify the vertices of the feasible region. The vertices are: \[ A(2, 3), \, B(3, 3), \, C(6, 0), \, D(2, 0). \] 
Step 3: Calculate the value of the objective function \( Z = 2x + 3y \) at each vertex.
\[ Z(2, 3) = 13, \quad Z(3, 3) = 15, \quad Z(6, 0) = 12, \quad Z(2, 0) = 4. \] 
Conclusion: The maximum value of \( Z \) is \( 15 \), achieved at the point \( (3, 3) \).