Contextualization

The linear system is a collection of linear equations, essential in the study of mathematics, as well as in solving practical problems of everyday life and in various areas of science. Basically, a linear system is a composition of two or more equations involving unknowns that we need to determine.

In its equations, the variable cannot be raised to a power greater than one and cannot be multiplied by another variable. For example, in the system of equations 2x+y=5 and 3x-2y=4, x and y are the unknowns that we need to find.

Within the study of linear systems, the discussion of the system is a fundamental topic. This discussion presents whether the system has a unique solution (called possible and determined), if it has no solution (called impossible), or if it has infinite solutions (classified as possible and undetermined). To identify in which of these situations the system is, we use Cramer's rule, which helps us verify the existence of solutions by performing a series of calculations.

Linear systems are not only present in math books, they are applied in various areas of knowledge, such as engineering, economics, physics, chemistry, among many others. For example, in civil engineering, linear systems are often used to calculate the balance of forces in a structure. In economics, linear systems help in making operational and strategic decisions, assisting in economic forecasts.

This topic is very relevant because the ability to solve linear systems and understand their solutions is an extremely valued skill in most professional areas involving mathematics and logic.

Students can use sources such as:

• Khan Academy - Several videos and exercises on linear systems are offered.
• Canal Matemática Rio - The YouTube channel Matemática Rio offers a series of explanatory and didactic videos on the subject.
• Brasil Escola - Provides theoretical content on the subject.

Practical Activity

Activity Title: "The Discussion of Systems in the Real World"

Project Objective:

This project will provide students with the opportunity to apply the concepts of linear systems and system discussion in a practical and realistic context. Students will work in groups of 3 to 5 people to solve real problems that require the application of linear systems and then discuss under which conditions the system is determined, undetermined, or impossible.

Detailed Project Description:

Students will be encouraged to identify a real-life problem where they can apply the concept of a linear system. It can be related to economics, engineering, physics, etc. They will formulate the problem in a way to derive a linear system from it and then discuss this system.

They will write a report describing in detail the process of problem identification, the formation of the linear system, the solutions found, and discussions about the system's conditions.

Students will have one month to complete the project.

Necessary Materials:

1. Notebook for recording ideas and resolutions.
2. Textbooks or internet access for research.
3. Computer with text processing software for the production of the final report.

Step by Step:

1. Group Formation: Students will divide into groups of 3 to 5 people.

2. Problem Selection: Groups must identify a real-life problem that can be represented and solved through a linear system. This phase should be documented, explaining why the chosen problem can be approached in this way.

3. Formation of the Linear System: Groups will transform the identified problem into a linear system, detailing all the steps of the conversion in their report.

4. System Resolution and Discussion: Groups will solve the formed linear system and discuss the system, verifying if it is determined, undetermined, or impossible.

5. Write the Report: Each group must produce a well-organized report describing the entire process, from problem identification to system resolution and discussion. The report should contain an Introduction, development (with the problem's contextualization, formation and resolution of the linear system, and the discussion of the system), conclusion, and the bibliography used.

Project Deliverables:

Groups must deliver:

• Written Report: The report must follow the defined format, containing: Introduction (with problem contextualization), Development (with detailed formation and resolution of the linear system and the discussion of the system), Conclusion (summarizing the main points, explaining the learnings obtained, and the conclusions drawn about the project), and the Bibliography.

The presentation of the concepts and applications of the linear system and the discussion of the system must be clearly evident in the report. The report should reflect not only the students' theoretical understanding of the concepts but also their ability to apply these concepts in practical scenarios.