Introduction

Relevance of the Topic

Linear Systems: Resolution is the beating heart within the body of Mathematics. It is a vital tool found in various everyday applications and in numerous professional fields, from engineering and computer science to physics and economics. More than just a manipulation of equations, linear systems represent real patterns, and solving these systems allows us to visualize, in a precise and tangible way, how these patterns interact with the real world. Mastering this topic is a gateway to a deeper and more complex understanding of Mathematics as a whole.

Contextualization

Linear Systems: Resolution is situated as an essential part of the general topic of linear equations and matrices. It is a crucial component not only for 3rd-year high school mathematics but also for higher education mathematics and beyond. It builds upon previous knowledge of linear equations, expanding this understanding to multiple variables and multiple equations working together. Furthermore, solving linear systems lays the groundwork for more complex and abstract studies of linear algebra, multivariable calculus, computer science, and mathematical modeling. Within the curriculum, this unit serves as an important bridge that connects prior knowledge to new, paving the way for more advanced mathematical concepts.

Theoretical Development

Components

• Linear Systems: A linear system is a collection of linear equations that share the same variables. These equations are often written in matrix form, and solving a linear system involves determining the values of the variables that make all the equations in the system true.

  • The linear system can be classified as consistent, inconsistent, or dependent. To be consistent, the system needs to have at least one solution. If it has none, it is called inconsistent. If there are infinite solutions, it is called a dependent system.

• Methods for Solving Linear Systems: There are three main methods for solving linear systems: substitution, elimination, and graphically. Each has its own advantages and disadvantages, and the choice of a method depends on the nature of the system and the solver's convenience.

  • The substitution method involves isolating a variable in one equation and substituting it into another equation. This process is repeated until the solution is obtained.

  • The elimination method aims to eliminate a variable by adding or subtracting the system's equations. This process is repeated until an equation with only one variable is reached. Thus, the solution can be obtained.

  • The graphical method plots the lines of each equation in the system on a Cartesian plane and identifies the point of intersection, which is the solution of the system.

• Key Terms:

  • Linear Equations: A linear equation is an equation that can be written in the form ax + by = c, where a, b, and c are constants and x and y are variables.
  • Matrix: A matrix is an ordered table of numbers arranged in rows and columns.
  • Augmented Matrix of a Linear System: A matrix obtained by adding the last column of constants to a matrix of coefficients.
  • Scaling: Scaling a matrix is the transformation of the matrix into a specific form, usually facilitating the resolution of systems of linear equations.

Examples and Cases

• Example of a Linear System Solved by the Substitution Method

  Let's consider the linear system:

  2x + y = 5
  x - y = 1
  

  Using the substitution method, we can isolate a variable in one equation and substitute it into the other. For example, if we isolate y in the first equation, we get y = 5 - 2x. Substituting y into the second equation, we get x - (5 - 2x) = 1, which simplified gives us x = 2. Now, substituting x into the first equation, we have 2(2) + y = 5, which gives us y = 1. Therefore, the solution to this system is x = 2 and y = 1.

• Case of a Linear System Solved by the Elimination Method

  Let's consider the following linear system:

  x + y = 4
  2x + 2y = 8
  

  In this case, we can see that the second equation is essentially a multiple of the first equation. Therefore, the system is dependent and has infinite solutions. Using the elimination method, if we multiply the first equation by 2, we will obtain the second equation. This confirms the dependence and tells us that for any value of x, the solution will be x + y = 4, which can be simplified to y = 4 - x. Therefore, this system has infinite solutions.

Detailed Summary

Key Points:

• Definition of Linear System: Understanding the concept of a linear system, which is a set of linear equations that share the same variables. Remembering that manipulating linear systems often involves rearranging these equations in matrix form.

• Classification of Linear Systems: Understanding the three categories in which linear systems can be classified: consistent, inconsistent, and dependent. Understanding that consistency refers to the presence of at least one solution, inconsistency to the absence of solutions, and dependence to the existence of an infinite number of solutions.

• Resolution Methods: Mastering the three main methods of solving linear systems - substitution, elimination, and graphical - and identifying which method is ideal for a particular system. Recognizing that each method has its own peculiarities, advantages, and limitations.

• Key Concepts: Identifying and understanding the terms intrinsic to linear systems, including linear equations, matrices, augmented matrix, and scaling. Understanding how these concepts relate to each other and facilitate the resolution of linear systems.

Conclusions:

• Reality Through Mathematics: We conclude that mathematics, specifically linear systems, allows us to analyze and understand structures and situations in the real world. The ability to discover the relationships and interactions between variables is fundamental in various disciplines and professions.

• Methods: Versatility and Choice: It is concluded that mastering various resolution methods is preferable, as it allows adaptability and the choice of the best approach for each situation. No method is superior to the other - each has its own advantages and disadvantages that depend on the system in question and the solver's preference.

Suggested Exercises:

1. Substitution Problem: Given the linear system below, solve it using the substitution method:

   2x + 3y = 7
   x - 2y = -1
   

2. Elimination Problem: Solve the following linear system by the elimination or addition/subtraction method:

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

3. Practical Problem: Imagine that you have a total of R$ 180 invested in two savings accounts, one with 2% annual interest and the other with 4% annual interest. At the end of a year, you received R$ 6 in interest. How much did you invest in each account? This is an example of an application problem of a system of linear equations, where the equations represent the total amount of money after a year in each account. Solve this linear system to find the amounts of money invested in each account.