Introduction

Relevance of the Topic

Studying the number of solutions of a system of linear equations is a fundamental topic in mathematics. It is a tool for solving real-world problems such as transportation planning, industrial production optimization, fuel blending, and many others. Its principles form the basis for more advanced concepts, such as vector space and linear transformation, which are widely used in computer science, engineering, and physics. Thus, this topic acts as a crucial building block for applied mathematics.

Contextualization

The study of systems of linear equations falls within the broad area of Algebra, which is a pillar of mathematics. Within Algebra, this topic is a natural extension of studies on individual linear equations. Instead of considering a single linear equation, we analyze a set of equations that share the same variables. The analysis of the number of solutions, therefore, deepens the understanding of the interaction between linear equations, variables, and solutions of a system. Furthermore, this content is linked, later on, to the introduction of the concept of a matrix, which is a powerful tool for solving systems.

Theoretical Development

Components

• System of Linear Equations: Consists of a set of linear equations with the same variables. Each equation is a constraint that must be satisfied by the variables of the system. For example, the system

  • 2x + 3y = 7
  • 4x + 6y = 14

  has two equations and two variables (x and y).

• Classification of Systems: Depending on the relationship between the number of equations (n) and the number of variables (m), a linear system can be classified as possible and determined, possible and indeterminate, or impossible.

  • Possible and Determined System (PDS): When the number of equations is equal to the number of variables and all equations are linearly independent.
  • Possible and Indeterminate System (PIS): When the number of equations is less than the number of variables, but the system has infinite solutions.
  • Impossible System (IS): When the number of equations is less than the number of variables and the system has no solutions.

• Criteria for Identifying the Classification: There are several ways to identify whether a system is PDS, PIS, or IS. Some important criteria are:

  • Criterion of Dependent Equations (CDE): If there is an equation that can be obtained as a linear combination of the others, the system is impossible.
  • Determinant Criterion (DC): If the determinant of the coefficient matrix is nonzero, the system is possible and determined.
  • Rank Criterion (RC): If the rank of the coefficient matrix is equal to the rank of the augmented matrix of the system, the system is possible and determined.

Key Terms

• Rank of a Matrix: It is the number of linearly independent rows (or columns) of the matrix. In the context of linear systems, the rank of the coefficient matrix and the rank of the augmented matrix play a fundamental role in the classification of the system.

• Determinant of a Matrix: It is a number associated with a square matrix. In the context of linear systems, the determinant of the coefficient matrix is used, along with the rank of the matrix, to classify the system.

• Known Space or Solution Space (KS/SS): This term refers to the set of all possible solutions of a linear system. In the case of a PDS, the SS is a single point. In the case of a PIS, the SS is a line, a plane, or a hyperplane in space. In the case of an IS, the KS is empty.

Examples and Cases

• Example 1 - Possible and Determined System (PDS): Consider the system of equations

  • 2x + 3y = 7
  • 4x + 5y = 14

  In this case, the number of equations (2) is equal to the number of variables (2) and the determinant of the coefficient matrix (2x5-4x3=2) is nonzero. Therefore, the system is classified as PDS, meaning it has a unique solution (x=1, y=2).

• Example 2 - Possible and Indeterminate System (PIS): In the system

  • 2x + 3y = 7
  • 4x + 6y = 14

  The number of equations (2) is less than the number of variables (2), but the determinant criterion does not apply, as the determinant is zero. Therefore, using the criterion of dependent equations, the system is classified as PIS, meaning it has infinite solutions.

• Example 3 - Impossible System (IS):

  In the system

  • 2x + 3y = 7
  • 4x + 6y = 15

  The number of equations (2) is less than the number of variables (2) and the determinant of the coefficient matrix (2x6-3x4=0) is zero. Therefore, the determinant criterion is applicable and the system is classified as IS, meaning it has no solution.

Detailed Summary

Key Points

• Identification of a System: The first crucial step is to recognize a collection of equations as a linear system. This identification occurs when there is a defined relationship between the equations, specifically when the equations share the same variables.

• System Classification: After understanding a system, it is essential to classify it correctly. This is done by considering the number of equations and variables, and applying the criteria of Possible and Determined System (PDS), Possible and Indeterminate System (PIS), or Impossible System (IS).

• Use of Criteria for Classification: The criteria of Dependent Equations (CDE), Determinant (DC), and Rank (RC) are essential analysis tools to determine the correct classification of a system. Each criterion has specific conditions that must be met for the classification to be confirmed.

• Association of Real Problem with the System: Defining the real problem in terms of linear equations, then identifying the real problem as a system, and finally, classifying the system according to the appropriate classification, is a practical aspect of studying the number of solutions of systems.

Conclusions

• Importance of the Concept: The number of solutions of a linear system is a fundamental characteristic of systems and has significant implications in applied mathematics. Therefore, the ability to identify and correctly determine the number of solutions of a system is an essential skill in algebra.

• Analysis Tools: The criteria of Dependent Equations (CDE), Determinant (DC), and Rank (RC) are powerful analysis tools that allow determining the correct classification of a system. Their correct use and full understanding are, therefore, fundamental.

• Connection with Applied Mathematics: Understanding the number of solutions of systems is a basis for many concepts in applied mathematics, such as optimization theory, linear programming, and many others. Therefore, this topic serves as a crucial building block for learning mathematics in a practical context.

Exercises

1. Exercise 1: Solve the following system:

   • 2x + 3y = 7
   • 4x + 5y = 14

   Identify the classification of the system and represent the solutions on the Cartesian plane.

2. Exercise 2: Consider the following system:

   • 2x + 3y = 7
   • 4x + 6y = 14

   Determine the classification of the system. Explain your reasoning, demonstrating why the system falls into that classification.

3. Exercise 3: For the system

   • 2x + 3y = 7
   • 4x + 6y = 15

   Apply the classification criteria (CDE, DC, RC) to determine the classification of the system. Justify your answer based on the criteria applied.