Contextualization

Mathematics is a tool that helps us solve complex day-to-day problems in a simplified and efficient way. In the universe of mathematics, solving systems of linear equations is a key concept for understanding various phenomena and situations in everyday life. In this project, we will learn how to use the Cramer's method to solve these systems of equations.

Cramer's method is a technique that consists of expressing each of the variables of a system of linear equations in terms of the determinants associated with the system. This method, due to its simplicity and its property of providing exact solutions, is used in a wide range of situations, from simple problem resolutions to engineering and computational sciences problems.

Solving the system by Cramer is not just a simple mathematical technique, but a powerful tool that allows us to understand and solve complex problems systematically and organized. By understanding and mastering the Cramer's method, we are able to address problems that, at first glance, seem impossible to solve.

Importance

Solving systems of linear equations is extremely relevant not only in the field of mathematics, but in various other areas of knowledge, from physics to economics. Through this method, it is possible to determine the behavior of several variables in terms of others, elucidating relationships and interactions that are fundamental to understand a wide variety of situations and problems.

Let's take a simple example of a linear system that represents the relationship between the price and quantity of two products. By solving the system, we can determine the quantity of each product that maximizes a company's profit. In the field of physics, systems of linear equations allow calculating variables such as speed, acceleration, and force from certain initial conditions.

Practical Activity

Activity Title: Unraveling the Cramer System in the Real World

Activity Objective

The objective of this activity is to apply Cramer's method to solve a system of linear equations that represents a real-world problem.

Activity Description

Each group of 3 to 5 students must identify a real-world problem that can be modeled by a system of linear equations, create this system, and solve it using Cramer's method.

Students can use everyday problems, economics, physics, engineering, or any other field that can be modeled using systems of linear equations.

Necessary Materials

1. Paper and pen for calculations and notes.
2. Computer with Internet access for research and use of calculation and graphic software, such as Desmos.
3. Books and other reference materials on Cramer's method and systems of linear equations.

Activity Step-by-Step

1. Students should initially research and choose a real-world problem that can be modeled by a system of linear equations. This problem can come from various areas, such as economics, physics, engineering, etc.
2. Next, the group must work together to create a system of linear equations that represents this situation.
3. With the system created, students must solve the system using Cramer's method, showing all calculation steps.
4. After solving the system, students must interpret the solutions in terms of the original problem. For example, if the system is about the production and price of two products, the solutions can represent the optimal quantity of each product to maximize profit.
5. Finally, the group must write a document detailing the entire process. This document should include an introduction to the problem and the choice of the system of equations, the description of the created system, the detailed resolution of the system using Cramer's method, the interpretation of the solutions, and conclusions about what the solutions mean for the original problem.

Project Delivery

At the end of the project, each group must deliver:

1. The written document containing:

   • Introduction: The group must introduce the chosen real-world problem and explain why a system of linear equations is appropriate for modeling this problem.
   • Development: The group must detail the system of equations they created, show step by step how they solved the system using Cramer's method, and interpret the solutions of the system in terms of the real-world problem.
   • Conclusion: The group must discuss what they learned from the project, how Cramer's method helped solve the system, and what the solutions of the system say about the real-world problem.
   • Bibliography: The group must indicate all sources they used to help in the creation and resolution of the system, as well as to understand and apply Cramer's method.

2. An oral presentation where each group member must explain a part of the project, demonstrating collaboration and understanding of the content by all.

The total time for completing this project is one week.