Introduction

Relevance of the Topic

The Linear System is the core of algebra, a fundamental pillar that permeates a wide range of mathematical and applied disciplines, including physics, economics, and engineering. The ability to understand and solve linear systems is a crucial skill for any Mathematics student. Moreover, it helps develop the ability to think logically, solve problems, and model mathematically.

Contextualization

In the Mathematics curriculum of the 3rd year of High School, the study of Linear Systems follows the topic of Matrices and determinants. After understanding these concepts, one dives into the study of systems of linear equations and methods to solve them.

A detailed discussion of a linear system plays a crucial role in building the student's understanding of the subject. In this context, the linear system is presented as a set of equations that work together to describe a situation or model. Through this unit, the "language" of Mathematics continues to become richer and more complex, paving the way for future topics such as linear algebra and calculus.

Studying the discussion of linear systems is an important step in deepening students' understanding of this topic. These concepts provide the foundation for understanding other topics in advanced mathematics, especially in calculus, linear algebra, and mathematical modeling courses.

Theoretical Development

Components

• Linear System: A linear system consists of a set of linear equations that share a common set of variables. It is compactly represented in matrix notation.

  For example, the linear system:

  2x + 3y = 8
  4x - y = 7
  

  Can be represented as:

  [2 3 | 8]
  [4 -1 | 7]
  

• Linear Equation: A linear equation is a first-degree polynomial equation. That is, it has the form ax + by + c = 0, where x and y are the variables of the equation and a, b, and c are real coefficients. This is the basic building block in the formation of a linear system.

• Independent Terms and Coefficients: Each term in a linear equation is either a coefficient (multiplied by a variable) or an independent term. In a system of linear equations, all coefficients and independent terms are organized into a matrix.

• Resolution Methods: There are different methods to solve a linear system, including the Elimination, Substitution, and Cramer methods. Each method uses matrix operations and linear equation operations to find variable values that satisfy all the system's equations.

Key Terms

• Discussion of Linear System: The discussion of a linear system occurs after translating the problem into the corresponding system of equations. The discussion focuses on classifying the system in terms of the number of solutions, which can be a possible and determined system (SPD), a possible and indeterminate system (SPI), or an impossible system (SI).

• Escalation: The process of transforming a linear system into a staggered form facilitates the resolution process and the identification of the discussion. The system will be in staggered form when the augmented matrix is reduced by rows.

• Matrix Form: The matrix form of a linear system represents the equations and unknowns using matrices. That is, the coefficients of the unknowns and the independent terms are organized into a matrix.

Examples and Cases

• SPD Example (Possible and Determined System): In the linear system

  [2 3 | 8]
  [4 -1 | 7]
  

  Given that |A| = 2, |B| = -12 and |C| = -3, that is, the determinant of the coefficient matrix is different from zero, then the system is possible and determined.

• SPI Example (Possible and Indeterminate System): In the linear system

  [2 4 | 8]
  [4 8 | 16] 
  

  All coefficients of the second equation can be expressed as multiples of the first, since the matrix A = [2 4; 4 8] is a matrix from which all columns are multiples of the second column. Therefore, the system is possible and indeterminate.

• SI Example (Impossible System): In the linear system

  [2 3 | 8] 
  [4 6 | 7]
  

  The coefficient of the first equation, when multiplied by 2 (coefficient of the second equation), does not result in the respective independent term. Therefore, the system is impossible.

Detailed Summary

Relevant Points

• Definition of Linear System: A linear system is a set of linear equations that share a common set of variables. They are represented compactly in matrix notation.

• Linear Equations: Each linear equation is a first-degree polynomial equation. In the context of systems, they are the basic building block in the formation of this.

• Resolution Methods: The teacher briefly addressed the methods of solving linear systems, including Elimination, Substitution, and Cramer. Each method uses specific techniques for manipulating matrices and equations to find the solutions of the system.

• Independent Terms and Coefficients: The teacher emphasized the importance of distinguishing independent terms from coefficients in a linear equation and in a system of linear equations.

• Discussion of Linear System: This is a crucial concept after translating the problem into the corresponding system of equations. The discussion focuses on classifying the system in terms of the number of solutions, and this classification is the core of the system discussion.

• Matrix Form: The teacher demonstrated how the representation of equations and unknowns in matrix form can facilitate the manipulation and resolution of systems.

• Representative Examples: The teacher presented representative examples of each type of system (SPD, SPI, SI), providing a clear understanding of how to identify and differentiate between them.

Conclusions

• The Discussion of Linear System is a crucial step after translating the problem into the corresponding system of equations, as it is at this stage that the system is classified in terms of the number of solutions, allowing understanding of the nature of the system.

• Understanding the Independent Terms and Coefficients, as well as their distinction, is vital for the resolution and discussion of linear systems.

• The Matrix Form of a linear system is a powerful tool for the resolution and manipulation of systems, providing a new approach to system analysis.

• The Resolution Methods presented offer the student various approaches to solving linear systems, each with its own advantages and disadvantages.

Suggested Exercises

1. Identification of Discussion: Given the linear system in matrix form, [3 4 | 7] [9 2 | 6], classify the system in terms of the number of solutions and justify your answer.

2. Resolution of System: Solve the following system of linear equations using the Substitution method:

   3x + 4y = 7
   9x + 2y = 6
   

3. Conversion to Matrix Form: Convert the following system of linear equations to matrix form:

   2x - 3y = 4
   5x + 6y = -1
   

4. Application of Resolution Methods: Solve the following system of linear equations using the Elimination method:

   2x - y + 3z = 4
   x + 2y - z = 1
   3x - 2y + 5z = 6