Lesson Notes By Weeks and Term v4 - SHS 2

Lesson Video

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Performance objectives

• Define feasible regions, vertices, and objective functions.
• Identify the vertices of a feasible region from a graph.
• Calculate the maximum and minimum values of a linear expression within a feasible region.
• Apply these concepts to solve problems.

Lesson summary

This lesson introduces a powerful application of algebra known as Linear Programming. At its core, this topic is about making the best possible decisions with limited resources. Imagine a small business owner in Kejetia Market who wants to make the most profit but only has a certain amount of starting capital and storage space. Or a farmer in the Volta Region deciding how much land to allocate to maize and cassava to get the best harvest, given a limited supply of fertilizer. Linear programming provides a mathematical way to solve these real-world problems. We will learn how to find the "best" (maximum or minimum) outcome by testing the corners of a region defined by our limitations.

Lesson notes

This topic, often called Optimization, is a part of a wider mathematical field called Linear Programming. The fundamental idea is to find the best possible value (e.g., maximum profit or minimum cost) for a situation described by a set of rules. Key Terminology Constraints: These are the limitations or rules of the problem, expressed as linear inequalities. For example, `x + y ≤ 100` could mean the total number of items produced (`x` and `y`) cannot exceed 100. Feasible Region: This is the area on a graph that contains all the possible solutions that satisfy *all* the constraints at the same time. It is typically a polygon. Any point `(x, y)` inside or on the boundary of this region is a "feasible" solution. Objective Function: This is a linear expression (e.g., `P = 5x + 7y`) that we want to maximize or minimize. 'P' could represent Profit, 'C' could represent Cost, etc. Our goal is to find the `(x, y)` point in the feasible region that gives the best possible value for this function. Vertices (or Corner Points): These are the points where the boundary lines of the feasible region intersect. The Corner Point Theorem This is the most important principle for this topic:

> The maximum or minimum value of an objective function will always occur at one of the vertices (corner points) of the feasible region.

This theorem simplifies our work tremendously. Instead of checking an infinite number of points inside the region, we only need to check the few corner points. Step-by-Step Method for Finding Maximum/Minimum Values Identify the Vertices: Carefully read the coordinates of all the corner points of the given feasible region from the graph. State the Objective Function: Write down the function you are asked to maximize or minimize. Test Each Vertex: Substitute the `x` and `y` values of each vertex into the objective function. Calculate the Results: Compute the value of the objective function for each vertex. It is helpful to organize this in a table. Compare and Conclude: Look at all the calculated values. The largest value is the maximum, and the smallest value is the minimum. State your answer clearly, including the value and the coordinates of the vertex where it occurs.

Evaluation guide

• Determine the maximum/minimum values within the given constraints.
• Collaborative Learning, Experiential Learning, Problem -based Learning, Project -Based
• Learning and Talk for Learning
• Activity: Feasible regions of systems of linear inequalities.