Applying First-Degree Equations in the Real World

Objectives

1. Recognize and solve first-degree equations.

2. Solve problems by setting up first-degree equations based on the data from the statement.

Contextualization

First-degree equations are fundamental in solving everyday problems. Imagine you are buying materials to build something and need to calculate how many items fit in your budget. Or consider a situation where you need to find out the average speed of a car to arrive on time for an appointment. These are practical applications of first-degree equations, which are essential tools for making quick and accurate decisions.

Relevance of the Theme

Knowledge of first-degree equations is crucial in the current context, as various professional fields such as engineering, finance, and logistics use these equations to optimize processes and make informed decisions. Moreover, the ability to solve first-degree equations is a practical skill that can be applied in various everyday situations, from financial planning to solving technical problems.

Concept of First-Degree Equations

A first-degree equation is a mathematical expression that represents an equality between two expressions, where the variable has a maximum degree of 1. This means that the variable is not raised to a power greater than one. These equations are fundamental for solving problems that involve linear relationships.

• First-degree equations have the general form ax + b = c, where a, b, and c are real numbers and x is the variable.

• The goal is to find the value of the variable that makes the equation true.

• These equations often appear in everyday and professional problems, being essential for various calculations and analyses.

Identification of Equation Components

The main components of a first-degree equation are the coefficients, constant terms, and variables. Understanding these components is crucial for the proper resolution of equations.

• Coefficient: The number that multiplies the variable (e.g., in 3x, 3 is the coefficient).

• Constant term: A number that does not vary and is not associated with a variable (e.g., in 3x + 5 = 20, 5 and 20 are constant terms).

• Variable: The unknown value we are trying to determine (e.g., x in 3x + 5 = 20).

Methods of Solving First-Degree Equations

There are different methods to solve first-degree equations, including simplification, isolation of the variable, and solution verification. These methods help find the value of the variable efficiently and accurately.

• Simplification: Combine like terms and simplify both sides of the equation.

• Isolation of the variable: Manipulate the equation to leave the variable alone on one side of the equality.

• Verification: Substitute the solution found back into the original equation to check if it satisfies the equality.

Practical Applications

• Engineering: Calculate the force needed to support a structure, where the force is directly proportional to the applied load.
• Finance: Determine the profit or loss of a company by analyzing fixed and variable costs and revenues.
• Logistics: Optimize the distribution of products by calculating the ideal quantity of items to be transported to minimize costs and maximize efficiency.

Key Terms

• First-Degree Equation: A mathematical equality where the variable has a maximum degree of 1.

• Coefficient: The number that multiplies the variable in an equation.

• Constant Term: A number that is not associated with a variable and does not change.

• Variable: The unknown value we are trying to determine in an equation.

Questions

• How can first-degree equations be applied in your daily life to solve financial or logistical problems?

• What advantages do you see in mastering the resolution of first-degree equations for your future professional career, regardless of the field?

• In what ways can understanding first-degree equations enhance your decision-making and problem-solving skills in complex situations?

Conclusion

To Reflect

Throughout this lesson, we explored the importance of first-degree equations and how they are applied in various everyday and workplace situations. From calculating the budget for an event to optimizing processes in companies, first-degree equations are indispensable tools for problem-solving and decision-making. Developing the skill to recognize and solve these equations not only enhances academic performance but also prepares students for practical challenges they will face in their future careers. Reflecting on these practical applications is essential to understand the relevance of the content learned and how it can be used to solve real issues.

Mini Challenge - Practical Challenge: Planning a Budget

In this mini-challenge, you will apply your knowledge of first-degree equations to plan the budget for a school event. Your goal is to determine the quantity of materials that can be purchased without exceeding the available budget.

• Form a group of 3 to 4 students.
• Read the problem statement: 'You need to buy posters, meters of fabric, and paint for a school event. Each poster costs R$ 10, each meter of fabric costs R$ 5, and each can of paint costs R$ 15. Your total budget is R$ 200. How many posters, meters of fabric, and cans of paint can you buy without exceeding the budget?'
• Identify the variables and formulate the corresponding first-degree equation.
• Solve the equation to determine the quantity of each item that can be purchased.
• Present your solution to the class, explaining the reasoning used to solve the problem.