Contextualization

Linear systems represent a set of linear equations that are related to each other. They can be used to solve a range of real-world problems, where the solution needs to consider several variables that influence each other. Mastering the concepts associated with these systems, their proper use, and the ability to discuss their results are fundamental skills in many areas such as physics, chemistry, economics, and computer science.

The discussion of the linear system involves analyzing the system to find out if it admits a solution, and if so, how many. We will examine three categories: possible and determined systems (which have a single solution), possible and undetermined systems (which have infinite solutions), and impossible systems (which have no solution).

Project Introduction

In our daily lives, we are surrounded by situations that can be described by linear systems. For example, in agriculture, a linear system can be used to maximize the yield of a crop considering different variables such as water, fertilizers, and planting area.

In economics, they are used to optimize production and profit considering variables such as production cost, market price, and demand. By understanding and applying the concepts of linear systems, you will be acquiring a powerful tool to solve complex problems in a simplified way.

Practical Activity: "The Balloon Company: Discussing Linear Systems in the Real World"

Project Objective

Learn, understand, and apply the concepts of linear systems and their discussion, working collaboratively to solve a practical problem.

Project Description

You will be part of the team of a fictional business: a company that sells balloons for events. The team will have to deal with a cost and resource optimization problem.

The company sells two types of balloon packages, A and B. Each package A contains 5 red balloons, 4 blue balloons, and 3 green balloons. Each package B contains 1 red balloon, 2 blue balloons, and 7 green balloons. Each red balloon costs R$2.00, blue costs R$1.50, and green costs R$1.00. The company has a fixed budget to be determined by the teacher.

The group's task is to determine how many packages of each type the company should produce to maximize the number of balloons, considering the available budget. The group should discuss the resulting system and present their conclusions.

Necessary Materials

1. Paper and pencil
2. Calculator
3. Internet or bibliographic resources for research.

Step by Step

1. Start by researching and reviewing the concepts of linear systems and their discussion.
2. Next, formulate the company's problem as a linear system.
3. Solve the system and discuss: Does it have a single solution (possible and determined), multiple solutions (possible and undetermined), or no solution (impossible)?
4. If there is a solution, how many packages of each type should the company produce?

Duration: This project should be completed in one week, and each student should expect to spend between 2 to 4 hours on it.

Group Size: Groups of 3 to 5 students.

Project Delivery

• Practical Part: Presentation in the classroom, each group must explain the formation of the system, the attempts to solve it, and the discussion of the system.

• Written Part: Each group must submit a report containing:

  1. Introduction: Provide context for the theme, its relevance, real-world application, the project's objective, and the description of the proposed problem.

  2. Development: Explain the theory behind the theme. Detail the process of solving the problem, from the formation of the system, the methodology used to solve it, to the discussion of the system. Present and discuss the results obtained.

  3. Conclusion: Summarize the main points of the work. Describe the learnings obtained and the conclusions drawn from the project.

  4. Bibliography: Indicate the sources used to work on the project such as books, web pages, videos, etc.

You will be evaluated both for technical accuracy and for the effectiveness of the presentation and collaboration within the group. Use this opportunity to delve into the subject and learn to work as a team. Good luck!