Grade 12 Mathematics Unit 4: Introduction to Linear Programming

Unit 4, “Introduction to Linear Programming,” introduces you to a powerful mathematical technique used for optimization in Grade 12 Mathematics. This unit focuses on how to make the best possible decisions given a set of constraints, a skill that is highly valuable in business, economics, and logistics.

Chapter 4.1 – Graphical Solutions of System of Linear Inequalities

This chapter builds the visual foundation for solving linear programming problems.

• Learn how to graph a system of linear inequalities on a coordinate plane.
• Define the feasible region as the overlapping shaded area that represents all possible solutions satisfying every inequality in the system.
• Identify the vertices (corner points) of the feasible region. These points are critical for finding the optimal solution.

Chapter 4.2 – Maximum and Minimum Values

This section introduces the core method of linear programming for finding the best outcome.

• Define the key components of a linear programming problem:
  • Objective Function: A linear equation representing the quantity you want to maximize (like profit) or minimize (like cost).
  • Constraints: The system of linear inequalities that define the feasible region.
• Understand the Vertex Theorem (or Corner Point Principle), which states that the optimal solution (maximum or minimum) will always occur at one of the vertices of the feasible region.
• Learn the step-by-step process for solving:
  1. Graph the constraints to determine the feasible region and its vertices.
  2. Substitute the coordinates of each vertex into the objective function.
  3. Compare the results to identify the maximum or minimum value.
• Explore how to use spreadsheet software to efficiently organize and solve linear programming problems.

Chapter 4.3 – Applications

Apply your new skills to solve real-world optimization problems.

• Learn how to translate word problems into a mathematical model by defining variables, writing an objective function, and setting up the constraints.
• Solve practical problems in various fields:
  • Business: Maximizing profit from producing different products with limited resources.
  • Manufacturing: Minimizing the cost of production while meeting demand.
  • Nutrition: Creating a diet plan that meets nutritional requirements at the lowest possible cost.

Learning Outcomes

By the end of this unit, you will be able to:

• Graph systems of linear inequalities and identify the solution regions.
• Construct linear programming models from given problems.
• Solve real-life problems using linear programming techniques.
• Apply the Vertex Theorem to find the maximum and minimum values of an objective function.
• Use spreadsheet software to solve linear programming problems.