Contextualization

Linear systems are fundamental in solving real-world problems in various areas such as engineering, economics, physics, among others. Understanding how to write them using matrices is an essential skill in learning mathematics and will deepen your knowledge of how these concepts apply in practical scenarios.

Theoretical Concepts

In mathematics, a linear system is a collection of one or more linear equations involving the same variables. For example,

x + y = 10
2x + 3y = 20

is a system of two equations in the two variables x and y. This system of linear equations can be more compactly represented using matrices.

Defining:

A = [[1,1], [2,3]], X = [[x],[y]] and B = [[10],[20]],

we can represent the system as AX = B.

This compact format is very useful when performing linear algebra operations, such as solving systems of equations.

Real-World Application

The ability to model real-world problems in mathematical terms is a valuable skill in various professions. Engineers, for example, often use systems of linear equations to model physical systems, such as electrical circuits and force systems. Economists use them to model economic systems, such as supply and demand. In the field of computing, they are used in optimization algorithms, image processing, neural networks, and many others.

Practical Activity

Activity Title: Solving Linear Puzzles

Project Objective

• Deepen the understanding of linear systems and their representation in matrix form.
• Develop teamwork, communication, time management, and problem-solving skills.

Project Description

In this activity, groups will be presented with a puzzle that can be solved through linear systems. They will have to convert the puzzle into a system of linear equations, write the system in matrix form, and then solve the system to uncover the puzzle's solution.

Required Materials

• Paper and pen for taking notes and solving problems.
• Calculator (optional).
• Books and websites for reference.

Step by Step

1. Form groups of 3 to 5 students.
2. Each group receives a different puzzle, which must be converted into a system of linear equations.
3. Each group should discuss and decide how to convert the puzzle into a system of linear equations. Take note of the reasoning steps.
4. Write the system of linear equations in matrix form.
5. Solve the matrix to find the solution to the system of linear equations and, consequently, unravel the puzzle.
6. Each group must prepare a presentation detailing the entire process, from reading the puzzle to solving the linear system. Include any difficulties encountered and how they were overcome.
7. Finally, write a detailed report following the guidelines below.

Project Delivery

Groups must deliver:

• Written Report: Presenting the puzzle's resolution, the transformation of the puzzle into a system of linear equations, the matrix representation, the system's solution, and the interpretation of the solution in the puzzle's context. The report should include:

  1. Introduction: Provide context to the problem, explaining the puzzle and its transformation into a system of linear equations.
  2. Development: Present the matrix representation of the system, the methodology used to solve the matrix, and discussion of the results. Cite and explain the theoretical concepts used and make a clear connection between theory and practice.
  3. Conclusion: Summarize the main points, explain what was learned, and draw conclusions about the project.
  4. Bibliography: List all sources used in the project.

• Oral Presentation: Each group will have to present to the teacher and the rest of the class how they solved the puzzle. It is important that all group members participate.

This activity should be completed in one week, and it is estimated that each student will take between two to four hours to complete it.