Contextualization

The study of systems of equations is one of the fundamental bases of Mathematics and has applications in various scenarios of our daily lives. Through the study of these systems, we can develop logical reasoning, improve problem-solving skills, and better understand situations that involve multiple variables and conditions.

In many situations, we come across problems that involve more than one variable and more than one condition. One way to solve these problems is through the use of systems of first-degree equations. Systems of equations are a set of two or more equations, where each one has two or more unknowns. The solution to these systems is the relationship of values that simultaneously satisfy all the equations of the system.

Understanding how many solutions a system can have is a fundamental aspect to know how to approach it and to understand what the solutions represent. A system can have a single solution (that is, a single set of values that satisfies all the equations), infinite solutions (that is, many different sets of values that satisfy all the equations), or no solution (no set of values that satisfy all the equations).

Importance

The applications of systems of equations in the real world are vast. They are used to model and solve problems in various areas such as physics, engineering, economics, biology, among others. In physics, for example, they can be used to calculate speeds and distances in motion problems. In economics, they can be used to optimize production and profit in situations involving multiple products or services.

In addition to being an essential topic in mathematics, the study of systems of equations is an excellent way to develop skills such as problem-solving, logical and abstract thinking. Understanding the concept of systems of equations and the number of solutions a system can have is fundamental to the formation of good mathematical thinking.

Practical Activity

Activity Title: "Solving the Puzzle of the System"

Project Objective

The objective of this project is to apply and deepen the knowledge about systems of first-degree equations with two unknowns and identify the number of solutions a system can have.

Detailed Project Description

The groups of students must create a fictional scenario of an everyday problem that can be represented by a system of first-degree equations with two unknowns. Then, they must create a visual presentation or a graphical representation of the problem using the Cartesian plane. Finally, they must solve the system and verify the number of solutions it has.

Necessary Materials

• Paper, pencil, eraser;
• Material for visual presentation (can be cardboard, flipchart, or presentation software like PowerPoint, Google Slides, etc.);
• Computer with internet access for research;

Detailed Step-by-Step for Carrying Out the Activity

1. Form groups of 3 to 5 students.

2. Hold a brainstorming meeting and discuss possible everyday scenarios that can be represented by a system of equations. Remember, the scenario should involve two unknowns.

3. After deciding on the scenario, create the representation of the system that models it. Write the equations that make up the system and clearly identify what each variable represents.

4. Use the substitution method or the comparison method, as appropriate, to solve the system. Identify if the system has a single solution, infinite solutions, or no solution.

5. Draw or create a visual presentation that illustrates the problem, the system's equations, the system's resolution, and the discussion of the number of its solutions. If possible, use a graph on a Cartesian plane to show the lines that represent the equations and their solutions.

6. Conclude the activity with a group discussion on what each solution of the system represents and how this understanding helps solve the proposed problem.

Project Deliverables

1. Creation and Resolution of the System: Students must present the created scenario and the equations of the system that represents it. They must show the resolution of the system and identify the number of its solutions.

2. Visual Presentation: Students must create a visual presentation (can be a cardboard, a flipchart, a PowerPoint presentation, Google Slides, etc.), showing the system, its resolution, and discussion of the number of its solutions.

3. Written Document: Students must write a report with the following topics:

   • Introduction: Contextualizing the theme and the relevance of the project.

   • Development: Presenting the chosen scenario, explaining the equations of the created system, detailing the methodology used to solve the system, and highlighting the number of solutions of the system. It should also explain how the number of solutions was identified and what each of them represents in the context of the problem.

   • Conclusions: Summarizing the main points of the work, the learning obtained, and the conclusions about the project. They should emphasize the application of systems of equations in the real world and the importance of verifying the number of solutions of a system.

   • Bibliography: Indicating the sources that were used for research and project execution.