Question

Solve the following linear programming problem graphically: Maximise \( Z = 20x + 30y \) Subject to the constraints: \[ x + y \leq 0, \quad 2x + 3y \geq 100, \quad x \geq 14, \quad y \geq 14. \]

Hint:

To solve linear programming problems graphically, first plot all the constraints and identify the feasible region. Then, evaluate the objective function at the vertices of the region to find the optimal solution.

Answer 1

This is a linear programming problem with the objective function \( Z = 20x + 30y \) and constraints. Step 1: Plot the constraints
We start by plotting the constraints on the graph:
1. \( x + y \leq 0 \): The line is \( x + y = 0 \), which intersects the axes at \( x = 0 \) and \( y = 0 \).
2. \( 2x + 3y \geq 100 \): The line is \( 2x + 3y = 100 \), which intersects the axes at \( x = 50 \) and \( y = 33.33 \).
3. \( x \geq 14 \): A vertical line at \( x = 14 \).
4. \( y \geq 14 \): A horizontal line at \( y = 14 \).
Step 2: Identify the feasible region The feasible region is the area that satisfies all the constraints. The region is bounded by the lines formed by the constraints.
Step 3: Evaluate the objective function Once we have the feasible region, evaluate \( Z = 20x + 30y \) at the vertices of the feasible region. The maximum value of \( Z \) will occur at one of these vertices.
Step 4: Find the optimal solution After evaluating \( Z \) at the vertices, the point where \( Z \) is maximized gives the optimal solution.