Lesson Notes By Weeks and Term v4 - SHS 2

Lesson Video

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Performance objectives

• Formulate linear inequalities from word problems.
• Graph inequalities to find a feasible region.
• Solve optimization problems using linear programming.
• Interpret solutions in real-life contexts.

Lesson summary

In our daily lives in Ghana, from the market woman in Makola deciding how much stock to buy, to the farmer in the Ashanti Region planning which crops to plant, people constantly make decisions to get the best possible outcome with limited resources. How can you run a business to make the most profit? How can a hospital provide the best care with a limited budget? Mathematics, specifically an area called Linear Programming, gives us a powerful tool to answer these questions. It is an application of algebra that helps in optimal resource allocation. Today, we will learn how to model these real-life situations using linear inequalities and solve them to make the best decisions.

Lesson notes

This topic, formally known as Linear Programming, is a mathematical method for finding the best possible outcome (e.g., maximum profit or minimum cost) in a given mathematical model for some list of requirements represented as linear relationships.

Let's break down the process into clear steps.

The Steps of Linear Programming Define the Decision Variables: These are the quantities you need to find. We usually represent them with variables like `x` and `y`. For example, `x` could be the number of bags of maize to produce, and `y` the number of bags of beans. Formulate the Objective Function: This is a linear equation that represents the quantity you want to maximise (like profit) or minimise (like cost). It will look like `P = ax + by` (for profit) or `C = ax + by` (for cost). Formulate the Constraints: These are linear inequalities that represent the limitations or restrictions in the problem (e.g., limited resources, time, or materials). They look like `cx + dy ≤ e` or `cx + dy ≥ f`. Include Non-Negativity Constraints: In most real-world problems, the variables cannot be negative. You cannot produce -5 shirts. So, we always include the constraints `x ≥ 0` and `y ≥ 0`. Graph the Inequalities: Draw a Cartesian plane and graph all the constraint inequalities. The region where all the shaded areas overlap is called the Feasible Region. This region represents all possible solutions to the problem. Identify the Vertices: The feasible region will be a polygon. The corners of this polygon are called vertices. These points are crucial because the optimal solution (maximum or minimum) will always occur at one of these vertices. You find them by finding the points of intersection of the boundary lines. Test the Vertices: Substitute the coordinates of each vertex into the objective function. State the Conclusion: The vertex that gives the largest value is the maximum, and the one that gives the smallest value is the minimum. Answer the original question in a complete sentence.

Evaluation guide

• Solve real life problems involving linear inequalities.
• Collaborative Learning, Experiential Learning, Problem -based Learning, Project -Based
• Learning and Talk for Learning
• Activity: Real -life applications of linear programming.