Context

Welcome to the Mathematics Challenge of the System of Equations. This is a powerful tool that allows us to solve various types of real-world problems. Let's understand what an equation and a system of equations are.

An equation is a mathematical sentence formed by two expressions separated by an equal sign. The word equation comes from the Latin aequatio, which means "to make equal". Therefore, an equation is nothing more than stating the equality between two expressions. For example, if we have the equation 5x = 15, we are stating that 5 times some number x is equal to 15. When we know that this number is 3, we say that we have solved this equation.

Now, a system of equations is a set of two or more equations with the same variables. When we solve a system of equations, we are finding the values for the variables that satisfy all the equations at the same time. See the system of equations below:

x + y = 10
2x + y = 15

The solution to this system is x = 5 and y = 5. If you substitute x and y in the two equations with these values, you will see that both equations are satisfied.

Importance and Applications

Mathematics is a universal language. And systems of equations are like complex puzzles of this language that we need to solve. They not only help us solve everyday problems, but also have numerous applications in various areas, such as physics, engineering, economics, statistics, and even in art and music.

For example, a merchant can use a system of equations to decide how much of each product to buy to maximize their profit, taking into account the cost and demand of each item. An engineer can use a system of equations to determine the forces acting on a structure. A doctor can use a system of equations to calculate the correct dosage of medications based on a patient's weight and health condition.

Practical Activity

Activity Title: Unveiling equations through the lost treasure

Project Objective

The objective of this project is for students to apply their knowledge of systems of equations in a fun and practical way. They will create their own "treasure hunt" problem that will be solved through systems of equations. The challenge is to propose a problem that can be solved with a system of equations and then graphically represent the solution on the Cartesian plane.

Detailed Project Description

Students, in groups of 3 to 5, will create a treasure hunt story. This story should be constructed in such a way that the solution to finding the treasure is solved through a system of equations. Each group should develop their own problems and respective solutions.

The problem should be created in a way that requires at least two different systems of equations to solve with two variables. An example of a problem would be: "A treasure map indicates that the treasure is hidden in a location that satisfies two conditions: it is the point where twice the number of steps north plus the number of steps east is equal to 10 and where the number of steps north plus three times the number of steps east is equal to 9. Where is the treasure?".

Once the problem is created, students must solve the system of equations and graphically represent it on the Cartesian plane. Finally, they must verify if the solution to the system actually solves the proposed problem.

Required Materials:

1. Sheets of paper or notebooks to write the story, equations, and solve the systems.
2. Pencils and erasers.
3. Ruler and compass (or graphics software) to draw the Cartesian plane.
4. Computer or tablet with internet access for research.

Detailed Step-by-Step

1. In groups, students should discuss and create a treasure hunt story. Remember, the story must be constructed in such a way that the solution to finding the treasure can be solved by a system of equations.
2. The group should translate the story into a system of first-degree equations with two variables. Make sure the created system of equations is correct and has a unique solution.
3. Solve the system of equations. Verify if the solution solves the proposed problem.
4. Graphically represent the equations on the Cartesian plane and identify the solution.
5. Prepare a final report containing all project documentation as per the following guidelines.

Project Deliverables and Writing the Written Document

Students must deliver a final report containing:

1. Introduction: The introduction should contain the presentation of the problem in the form of a story. It should mention why they chose such a problem and the relevance of the system of equations to solve it. Additionally, the objective of this project should be mentioned.
2. Development: In this section, students should describe in detail the developed equations, the process of solving the system and graphical representation. They should also include discussions on verifying the solution and interpreting the result in the context of the problem.
3. Conclusions: Students should summarize the main points of the work, explicitly stating the learnings obtained, the difficulties faced, the solutions found to overcome these difficulties, and the conclusions drawn about the project.
4. Bibliography: References consulted should be informed, such as books, websites, videos, and other sources.

The report should be written clearly and objectively, using a formal and scientific language. The graphical representation can be done using an application or software and inserted in the report with the necessary explanations. It is recommended that students review the report before submission to avoid grammatical errors and ensure the clarity of the information presented.