Linear Programming and Systems of Linear Inequalities: Business Calculus Study Notes

Linear Programming

3.1 Setting Up Linear Programming Problems

Linear programming is a mathematical technique used to find the maximum or minimum value of a linear function, subject to a set of linear constraints. This section introduces the foundational concepts and steps for formulating linear programming problems.

• Linear Programming Problem: The process of finding an extreme value (maximum or minimum) of a linear function, subject to certain constraints.

• Objective Function: The function being optimized (maximized or minimized).

• Constraints: The conditions that must be satisfied, typically expressed as linear inequalities or equations.

Steps for Setting Up a Linear Programming Problem:

1. Identify and clearly define all variables.

2. Indicate and write the objective function.

3. Indicate and write the constraints.

Example: A food truck sells two types of sandwiches. Let x = number of Plain Jaine sandwiches, y = number of Sloppy Goddess sandwiches. The objective is to maximize revenue, subject to ingredient constraints and non-negativity (x ≥ 0, y ≥ 0).

Additional info: In many problems, non-negativity constraints (variables must be ≥ 0) are assumed even if not explicitly stated.

Graphing Systems of Linear Inequalities in Two Variables

3.2 Graphing Systems of Linear Inequalities

This section covers how to graph systems of linear inequalities in two variables, which is essential for visualizing feasible regions in linear programming problems.

• Linear Inequality: An expression such as $ax + by \\leq c$ or $ax + by \\geq c$.

• Solution Set: The set of all points (x, y) that satisfy the inequality.

• Boundary Line: The line corresponding to the equation $ax + by = c$.

Types of Linear Inequalities:

• Non-Strict: Includes the boundary line (e.g., $y \\leq 2x + 3$).

• Strict: Does not include the boundary line (e.g., $y < 2x + 3$).

Steps for Graphing a Linear Inequality:

1. Graph the boundary line $ax + by = c$ using x- and y-intercepts.

2. Determine if the boundary line is included (solid line for non-strict, dashed for strict).

3. Choose a test point (often (0,0) if not on the boundary).

4. Substitute the test point into the inequality to determine which side of the line is the solution set.

Example: Graph the solution set for $x + 3y \\leq 15$, $-2x + 4y \\geq 0$, $x \\geq 0$, $y \\geq 0$.

Bounded and Unbounded Solution Sets

The feasible region for a system of linear inequalities can be either bounded or unbounded.

• Bounded: The region can be completely enclosed by a circle.

• Unbounded: The region cannot be completely enclosed by a circle.

Corner Point: The point where the edge of the solution set changes from one boundary line to another. These are critical in linear programming, as optimal solutions often occur at corner points.

The Method of Corners

Solving Linear Programming Problems Graphically

The Method of Corners is a graphical technique for solving linear programming problems. It involves identifying the feasible region defined by the constraints and evaluating the objective function at each corner point of this region.

• Graph the constraints to find the feasible region.

• Identify the corner points (vertices) of the feasible region.

• Evaluate the objective function at each corner point.

• The maximum or minimum value of the objective function will occur at one of the corner points.

Example: Maximize $P = 4x + 3y$ subject to $2x + y \\leq 10$, $2x + 3y \\leq 18$, $x \\geq 0$, $y \\geq 0$.

Definition: The solution to a linear programming problem is the set of all points (x, y) that satisfy all constraints. The optimal value is found by evaluating the objective function at each corner point of the feasible region.

Summary Table: Key Concepts in Linear Programming

Term	Definition	Example
Objective Function	Function to be maximized or minimized	$P = 4x + 3y$
Constraint	Condition that must be satisfied	$2x + y \\leq 10$
Feasible Region	Set of all points satisfying all constraints	Shaded region on graph
Corner Point	Vertex of the feasible region	Intersection of two boundary lines
Bounded Region	Region enclosed by constraints	Polygonal area
Unbounded Region	Region not fully enclosed	Area extending infinitely

Practice and Application

Sample Problems and Reflection

Students are encouraged to practice setting up linear programming problems, graphing systems of inequalities, and identifying feasible regions and corner points. Reflection questions may include:

• How do you graph the solution set for a linear inequality or system of inequalities?

• How do you find the coordinates of corner points?

• What is the difference between bounded and unbounded solution sets?

Additional info: These skills are foundational for business calculus applications such as resource allocation, cost minimization, and profit maximization.