Lesson Notes By Weeks and Term v4 - SHS 2
APPLICATION OF ALGEBRA
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APPLICATION OF ALGEBRA
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Subject: Additional Mathematics
Class: SHS 2
Term: 1st Term
Week: 12
Grade code: 2.1.1.LI.9
Strand code: 1
Sub-strand code: 1
Content standard code: 2.1.1.CS.3
Indicator code: 2.1.1.LI.9
Theme: MODELLING WITH ALGEBRA
Subtheme: APPLICATION OF ALGEBRA
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In our daily lives in Ghana, from the market woman in Makola deciding how much cassava and plantain to stock, to the factory manager in Tema planning production, people constantly make decisions to get the best possible results with limited resources. Algebra, specifically the topic of Linear Programming, gives us a powerful mathematical tool to solve these kinds of problems. It helps us model real-life situations using linear inequalities to find the optimal solution, such as the maximum profit or the minimum cost. This lesson will equip you with the skills to translate everyday challenges into mathematical models and solve them systematically.
This topic, formally known as Linear Programming, is a method for finding the best outcome in a mathematical model whose requirements are represented by linear relationships. Key Terminology Decision Variables: These are the quantities you need to decide on. We usually represent them with variables like `x` and `y`. For example, `x` could be the number of bags of rice to produce, and `y` could be the number of bags of gari. Constraints: These are the limitations or restrictions in the problem, such as limited resources (money, materials, time), or demand. We express constraints as linear inequalities (e.g., `2x + 3y ≤ 100`). Objective Function: This is a linear equation that represents the quantity we want to maximize (like profit) or minimize (like cost). It is usually written in the form `P = ax + by` (for profit) or `C = ax + by` (for cost). Feasible Region: When we graph all the constraint inequalities, the area on the graph that satisfies ALL the constraints at the same time is called the feasible region. Any point within this region is a possible solution to the problem. Vertices (or Corner Points): These are the points where the boundary lines of the feasible region intersect. A fundamental principle of linear programming is that the optimal solution (maximum or minimum value) will always occur at one of these vertices. The Step-by-Step Method for Solving Linear Programming Problems Identify and Define Variables: Read the problem carefully and decide what quantities you are trying to determine. Assign variables (e.g., `x`, `y`) to them. Formulate the Constraints: Write down all the limitations as a system of linear inequalities. Don't forget the non-negativity constraints (i.e., `x ≥ 0` and `y ≥ 0`), because you cannot produce a negative number of items. Formulate the Objective Function: Write the equation for the quantity you want to maximize or minimize. Graph the Constraints: For each inequality, treat it as an equation to find the boundary line. Graph these lines on a Cartesian plane. A simple way is to find the x- and y-intercepts. Then, shade the region that satisfies the inequality. The overlapping shaded area is the feasible region. Identify the Vertices: Find the coordinates of all the corners (vertices) of the feasible region. Some vertices will be on the axes, while others are the intersection points of two boundary lines. To find intersection points, you must solve the two corresponding linear equations simultaneously. Test the Vertices: Substitute the coordinates of each vertex into the objective function. Determine the Optimal Solution: Compare the values obtained in the previous step. The largest value is the maximum, and the smallest value is the minimum. State your answer clearly in the context of the original problem.