# Introduction
### Relevance of the Topic
First-degree equations form the basic structure for many mathematical concepts that follow. If you have ever wondered what mathematics is for, here is an answer: to solve practical problems, but for that, we need to understand equations and how they operate. Mathematics, believe it or not, is a method of communication and equations are an essential grammar of that language.
### Contextualization
First-degree equations are the next step after learning basic arithmetic operations and the properties of equality. They appear at a crucial moment in the discipline of mathematics, providing the transition between simple arithmetic and the complexity of algebra. Within the mathematics curriculum, first-degree equations are essential for understanding algebraic structures and formulas, financial mathematics, geometry, and even the principles of physics and engineering.
This is a turning point in any student's journey in mathematics, where the ability to solve first-degree equations can unlock the understanding of more advanced topics and open doors to domains that were previously inaccessible.
# Theoretical Development
- **First-Degree Equation:** A first-degree equation, or linear equation, is represented in the form ax + b = c, where a, b, and c are real numbers and a ≠ 0.
- **a:** This is the coefficient of x, the variable term of the expression. It is the value multiplied by the unknown and determines the linear rate of change of the equation.
- **x:** The unknown, or variable, of the equation. The solution to the equation is the value of x that makes the equation true.
- **b:** This is the constant term of the expression, which represents the initial value of the equation, that is, when x = 0.
- **c:** This is the result of the equation, which is obtained after manipulating the left side of the equation.
- Importance: Solving first-degree equations is fundamental for making decisions and solving real-world problems. Examples include calculating salaries, discounts, interest rates, distances, and many others.
- **Key Terms:**
- **Coefficient:** In an equation, the numbers next to the variables are called coefficients. In the case of the first-degree equation, the coefficient is 'a'.
- **Variable:** The letters in an equation are called variables, representing the unknown values being solved. In the equation ax + b = c, x is the variable.
- **Constant Term:** The number that stands alone in an equation, without any associated variable, is called the constant term. In the case of the first-degree equation, the constant term is 'b'.
- **Result:** It is the value we obtain when solving the equation. In the equation ax + b = c, c is the result.
- **Examples and Cases:**
- **Example 1:** If a taxi ride charges a fixed fee of $5 (constant value) and then an additional $3 per kilometer traveled (variable term), the total price of the ride (result) can be represented by the equation 3x + 5 = c. Here, x is the number of kilometers traveled.
- **Example 2:** Suppose you want to buy a game that costs $60, but you have a 15% discount coupon on the game price. The equation representing this situation is 60 - 0.15x = c, where x is the original price of the game.
- **Example 3:** If you have savings and add $500 to it every month. The total savings after x months can be represented by the equation 500x = c.
- In all these cases, we form a first-degree equation representing a real-world situation and solve it to find the unknown variable.
# Detailed Summary
- **Key Points:**
- **Equation Structure:** Understanding the structure of the first-degree equation, represented in the form ax + b = c, is crucial. In this representation, a is the coefficient of the variable x, b is the constant term, and c is the result of the equation. This knowledge is the cornerstone for the correct understanding of first-degree equations.
- **Terminology:** Familiarity with terms such as coefficient, variable, constant term, and result is essential to understand and solve first-degree equations. This vocabulary is the language of mathematics and the basis for delving into more advanced concepts.
- **Practical Applications:** First-degree equations appear in many everyday situations. From calculating the cost of a taxi ride to understanding a discount on a purchase or the growth of savings - understanding first-degree equations allows us to quantify and make informed decisions in the real world.
- **Conclusions:**
- **Mastering Basic Mathematics:** The ability to solve first-degree equations is a significant milestone in any student's mathematical journey. This is the first contact with algebraic resolution that will lead to the understanding of more advanced topics.
- **Problem Solving:** First-degree equations are powerful tools for problem solving. They allow us to represent real-life scenarios in mathematical terms and then solve them effectively.
- **Abstraction and Concretization:** First-degree equations help develop the ability to abstract information, translating it into mathematical language, and then concretize it again when the equation is solved, obtaining the solution to the real situations they represented.
- **Exercises:**
- **Exercise 1:** A taxi driver charges a fixed fee of $10 plus $2 per kilometer traveled. How much will a 15-kilometer trip cost?
- **Exercise 2:** In a bookstore, all books have a 20% discount. If a book costs $50, what will be the price of the book after the discount?
- **Exercise 3:** You are saving to buy a new cell phone that costs $1200. Every month you save $100. How many months will it take for you to save the necessary money?
Each of these exercises was designed to test the understanding and practical application of first-degree equations, representing real scenarios that can be faced in our daily lives.
