Contextualization

In this project, we will explore the fascinating world of equations with two variables. Whether in economics, biology, physics, chemistry, or in numerous other areas, equations play a fundamental role in representing and solving complex problems. Equations with two variables, in particular, are of great importance because they allow us to describe relationships between two distinct quantities. For example, how temperature and pressure are related in an ideal gas, how the production cost of a product is related to its demand, among others.

Equations with two variables are a form of mathematical model that can be used to describe a wide variety of real-life situations. Models are tools that allow us to simplify reality in order to better understand the processes that are happening and make predictions. Theoretically, these equations express the relationship between two variables, such as x and y. If you know the value of one of the variables, you can find the value of the other using the equation. This is incredibly useful in a variety of contexts!

Introduction

Before we begin, let's clarify what a two-variable equation is. The equation with two variables is one that contains two unknowns, usually represented by x and y. Such equations express a relationship between these two variables. In the most common form, the linear equation with two variables (or first-degree equation with two variables) is written as ax + by = c, where a, b, and c are real numbers and a and b are not simultaneously equal to zero.

When we solve an equation like this, we are looking for an ordered pair (x, y) that satisfies the equation, that is, that makes the equality true. The solutions of a two-variable equation can be represented graphically on a Cartesian plane, where the solutions correspond to the points (x, y) on the line that the equation describes.

To delve even further into the concepts covered here, we suggest the following sources:

• Brazil School - First-degree equation with two variables
• Khan Academy - Systems of equations
• Simple Mathematics - First-degree equation with two variables

Practical Activity

Activity Title: "Expressing Real Life in Equations"

Activity Objective

Apply the concept of equations with two variables to solve problematic everyday situations, using a Cartesian plane to graphically represent the solutions of the equations.

Detailed Activity Description

Each group of students should choose a real-life situation that can be translated into a two-variable equation. For example, it can be the relationship between the number of study hours and the grade on a test, the relationship between speed and the time taken to travel a distance, among others.

Students should collect data related to the chosen situation (through research, questioning, observation, etc.) and use this data to form their equation. They should then solve the equation to find possible values for the two unknowns.

The results should be represented on the Cartesian plane. Finally, students should interpret the relationship between the variables based on the equation and the produced graph.

Necessary Materials

• Graph paper
• Pencils and erasers
• Colored pens
• Ruler
• Compass
• Computer with internet access for research

Activity Step by Step

1. Form groups of 3 to 5 students.

2. Choose a real-life situation that can be translated into a two-variable equation.

3. Research and collect data related to the chosen situation.

4. Use the collected data to formulate the equation that translates the chosen situation.

5. Solve the equation, finding possible values for the two unknowns.

6. Graphically represent the solutions of the equation on the Cartesian plane.

7. Interpret the relationship between the two variables based on the equation and the graph.

Project Deliverables

Practical Part

The practical part of the project will be an infographic, made on graph paper, which must contain:

• The chosen real-life situation.
• The two-variable equation that translates the situation.
• The solutions found for the unknowns.
• The graphical representation of the solutions on the Cartesian plane.
• The interpretation of the relationship between the variables.

Theoretical Part

The theoretical part of the project will consist of a report of at least two pages, containing:

1. Introduction: Students must introduce the chosen situation, explain why it can be translated into a two-variable equation and how this is relevant in real life. They must also present the project's objective.

2. Development: Students must describe the methodology used to collect the data, how they formulated the equation based on the collected data, how they solved the equation, and how they represented the solutions on the Cartesian plane. They must also present and discuss the results obtained, interpreting the relationship between the two variables.

3. Conclusion: Students must conclude the report by summarizing the main points, explaining the learnings obtained, and the conclusions drawn from the activity.

4. Bibliography: Students must list all sources of information consulted in the project.

The total duration of the project (including the practical part and the report writing) should be 2 to 4 hours per student, and the total delivery time is one week.