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. The objective function is \( Z = 20x + 30y \), subject to several constraints. Step 1: Graph the constraints. The constraints are plotted on a graph: 1. \( x + y \leq 0 \): The boundary line is \( x + y = 0 \). It intersects the axes at \( x = 0 \) and \( y = 0 \). 2. \( 2x + 3y \geq 100 \): The boundary line is \( 2x + 3y = 100 \). It intersects the axes at \( x = 50 \) and \( y = 33.33 \). 3. \( x \geq 14 \): This is a vertical line at \( x = 14 \). 4. \( y \geq 14 \): This is a horizontal line at \( y = 14 \). Step 2: Determine the feasible region. The feasible region is the area on the graph that satisfies all the given constraints. This region is enclosed by the lines derived from the constraints. Step 3: Calculate the objective function values. Evaluate the objective function \( Z = 20x + 30y \) at each vertex of the feasible region. The maximum value of \( Z \) will be found at one of these vertices. Step 4: Identify the optimal solution. The point (vertex) within the feasible region that yields the maximum value for \( Z \) represents the optimal solution.