Question

In a Linear Programming Program (LPP) for objective function \( Z = 14x - 10y \) subject to the constraints: \[ x + y \leq 8, \quad 3x - 2y \geq -6, \quad x, y \geq 0 \] shade the feasible region and mark the corner points in a neatly drawn graph.

Hint:

When solving a Linear Programming Problem, graph the constraints and identify the feasible region where all inequalities overlap. The objective function is then evaluated at the corner points.

Answer 1

To determine the solution for this Linear Programming Problem, we begin by plotting the constraints to identify the feasible region. 1. Plot the line \( x + y = 8 \), which can be expressed as \( y = 8 - x \).
2. Plot the line \( 3x - 2y = -6 \), which can be expressed as \( y = \frac{3x + 6}{2} \)
3. The inequalities \( x \geq 0 \) and \( y \geq 0 \) define the first quadrant.
The feasible region is the area of overlap for all inequalities, depicted as the yellow-shaded region in the graph. The vertices of the feasible region are the intersection points of the lines: - The point \( (0, 8) \) is the intersection of the line \( x + y = 8 \) with the y-axis.
- The point \( (2, 2) \) is the intersection of the lines \( x + y = 8 \) and \( 3x - 2y = -6 \).
- The point \( (8, 0) \) is the intersection of the line \( x + y = 8 \) with the x-axis.
These constitute the feasible corner points.
% Graph of the feasible region \includegraphics[width=0.7\textwidth]{output.png}