Lesson Notes By Weeks and Term v4 - SHS 2

Lesson Video

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Performance objectives

• Graph systems of linear inequalities.
• Determine the feasible region from a system of inequalities.
• Calculate the maximum and minimum values of a function within constraints.
• Apply these methods to solve real-life problems.

Lesson summary

This lesson introduces a powerful mathematical technique called Linear Programming. This is a method used to find the best possible outcome (like maximum profit or minimum cost) in a given situation with certain limitations or rules. In Ghana, from the market woman deciding how much stock to buy, to the farmer planning crop planting, to large companies managing production, people constantly face problems of how to best use limited resources. Linear programming gives us a systematic, algebraic and graphical way to solve these real-world problems.

Lesson notes

This topic, Linear Programming, is a method for optimization. It involves several key components:

a) Objective Function: This is a linear equation that represents the quantity you want to maximize or minimize. It's usually written in the form P = ax + by (for Profit) or C = ax + by (for Cost), where `x` and `y` are the variables you can control. Example: If a carpenter makes a profit of GH₵ 50 on a chair (`x`) and GH₵ 80 on a table (`y`), the objective function for profit is P = 50x + 80y.

b) Constraints: These are the limitations or restrictions in the problem, expressed as linear inequalities. They represent limited resources like time, materials, money, or space. Example: If the carpenter has only 40 hours of labour available, and a chair takes 2 hours while a table takes 5 hours, the constraint is 2x + 5y ≤ 40. Non-negativity Constraints: In most real-world problems, the variables cannot be negative. We can't make "-3 chairs". So, we almost always have the constraints x ≥ 0 and y ≥ 0.

c) Feasible Region: When we graph all the constraint inequalities on a Cartesian plane, the area that satisfies *all* the inequalities simultaneously is called the feasible region. This region contains all the possible solutions to the problem. Any point inside or on the boundary of this region is a "feasible" solution.

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.