Contextualization

Theoretical Introduction

Mathematics is an applied science that permeates various areas of our daily lives and is essential for understanding the world around us. Among the numerous concepts, one that deserves attention is the system of equations, which allows us to deal with multiple equations simultaneously.

In the simplest context, a system of equations is a set of two or more equations involving two or more unknowns. Its objective is to find a set of values that satisfy all the equations of the system at the same time. Thus, when working with systems of equations, we are seeking the intersection of the solutions of the equations that make up the system.

But an important point to emphasize is that this intersection does not always exist, or it can be a set of infinite solutions. In general, a system of equations can have a unique solution, infinite solutions, or no solution.

Relevance of the Theme

Systems of equations are applied in various areas such as physics, chemistry, economics, engineering, computer science, among others. Therefore, knowing how to solve a system of equations and determine the number of solutions it has is an essential skill for solving complex problems in these areas.

It is possible to use systems of equations to solve problems such as determining the quantity and cost of transporting goods on a commercial route, calculating equilibrium points in economics, or even in solving games and puzzles.

Practical Activity

Activity Title: Unraveling Systems of Equations in the Business World

Activity Objective:

The objective of this activity is to allow students to understand the theory of first-degree systems of equations and the number of solutions, and then apply these concepts to solve real-world problems, specifically in the business sector. This project will also aim to integrate mathematics and economics, stimulating students to develop their teamwork, time management, problem-solving, creative thinking, and proactivity skills.

Activity Description:

Groups will be asked to create a small fictional business, with products and prices of their choice. From this, they must create a series of scenarios involving these products, taking into account production costs, sales, and profits. For each scenario, they must formulate systems of equations that represent the business's economic situation at that moment.

They must solve these systems of equations manually and using mathematics software (such as GeoGebra or Desmos) and determine how many solutions exist for each of the systems, presenting the results graphically when possible.

Students should also investigate scenarios in which the system of equations has no solution or has infinite solutions and explain what these scenarios represent from a business perspective.

Required Materials:

• Paper and pencil for notes and calculations
• Computers with Internet access and mathematics software (GeoGebra or Desmos)
• Research material to better understand the economic variables involved in a business.

Step by Step:

1. Group Formation: Divide into groups of 3 to 5 students.

2. Business Creation: Develop a small fictional business, choosing products and setting prices.

3. Scenarios and Systems of Equations: For each product, create scenarios involving production costs, sales, and profits and establish the systems of equations that represent these scenarios.

4. Solving the Systems of Equations: Solve the systems of equations manually and using mathematics software. Determine the number of solutions for each system.

5. Graphical Representation: Graphically represent the solutions of the systems, whenever possible.

6. Scenario Analysis: Analyze the scenarios where the system of equations has no solution or has infinite solutions. Discuss what these scenarios represent in the business context.

7. Final Report: Prepare a report with the following topics: Introduction, Development, Conclusions, and Bibliography.

8. Project Presentation: Prepare a project presentation for the class, explaining the concept of your business, the created scenarios, how they were translated into systems of equations, the number of solutions for each system, and what they represent in the business context.

Project Deliverables:

Students must submit a written report of the project and make a presentation to the class.

The report should include:

1. Introduction: Description of the fictional business created by the group, and the relevance of mathematics and systems of equations in managing a business.

2. Development: Explain the theory of systems of equations, the resolution methodology, and the mathematics software used. It should also present each created scenario, the formulated equations, and the results obtained.

3. Conclusions: Students should discuss what they understood about the number of solutions of a system of equations and what these solutions represent in the business context. They should also reflect on what they learned about the role of mathematics in economics and business management.

4. Bibliography: List the research sources used during the project development.

The presentation to the class should summarize the main points of the report and include graphs that visually represent the solutions of the systems of equations.