02_Ex 5.2

Exercise Questions

Questions	Links
Q1. Graph the feasible region of the following…	5.2 Q-1
Q2 Graph the feasible region of the following..	5.2 Q-2

The Questions provide a general method and might skip over specific parts, Refer to video appended at the bottom for a deeper understanding of each part

Overview

This exercise involves solving systems of linear inequalities by graphing the feasible region and identifying the corner points of the region. Understanding the graphical representation of these systems is essential for solving linear programming problems, particularly in optimization and resource allocation.

Key Concepts

Feasible Region: The set of all possible solutions to a system of inequalities that satisfies all constraints simultaneously.
Corner Points: Points where the boundary lines of the feasible region intersect. These points are crucial for solving linear optimization problems.
Linear Inequalities: Mathematical expressions involving linear terms, where solutions must satisfy the inequality relationship (e.g., $2x - 3y \leq 6$ ).
Graphing System of Inequalities: The process of plotting inequalities on a coordinate plane and identifying the region where all inequalities overlap.

Important Formulas

Linear Inequality: $ax + by \leq c$ or $ax + by \geq c$
Intersection of Two Lines: Solve the system of equations by substitution or elimination to find the coordinates of the intersection point.
Test Point Method: Use specific values (like $x = 0$ or $y = 0$ ) to determine the feasible region.

Tips and Tricks

Test points on the boundary: To determine which side of the line the inequality holds, substitute the point into the inequality.
Graphically identify feasible regions: The feasible region is where all inequalities overlap, often forming a polygon.
Corner points: Always check the corner points of the feasible region, as these are often the solutions to optimization problems.

Summary

In this exercise, you are tasked with graphing the feasible regions for different systems of linear inequalities. Each inequality defines a boundary, and the feasible region is where all the inequalities overlap. After graphing, the corner points are found, which are critical for optimization. The solution to each system involves identifying these points of intersection in the first quadrant, where the inequalities are satisfied.

Reference

By Sir Shahzad Sair: