Introduction

Relevance of the Topic

Within the vast field of Mathematics, solving first-degree equations problems holds a crucial position. Studying these problems provides the foundation for understanding more advanced concepts, such as solving second-degree equations and manipulating algebraic expressions. Not only that, but it is also a practical skill that can be applied in everyday situations, such as calculating discounts on purchases or sizing up revenues.

Contextualization

"In the 1st year of High School, Mathematics is at a point of transition between conceptual calculation and the application of mathematical knowledge in real-world problems. Solving first-degree equations problems lies at the heart of this transition. Gaining depth in the understanding of variables, students begin to deal with the complexity that linear equations can present. This topic is a seed planted for the future growth of mathematical skills and must be successfully mastered for curriculum progress. This is where the bricks are laid for the future construction of more advanced skills in Calculus, Linear Algebra, and many other related disciplines. Therefore, mastering this topic is not only essential for the study of Mathematics as a whole but also for the development of a solid logical and deductive reasoning."

Theoretical Development

Components

• First-Degree Equations: First-degree equations, also known as linear equations, have the general form ax + b = c, where a, b, and c are real numbers and a ≠ 0. In this type of equation, the variable x is raised to the first power (degree one), hence the name "first-degree equation".

• Terms of the Equation: In the expression ax + b = c, the elements a, x, b, and c have specific roles. a is the coefficient of the variable x, b is the independent term, and c is the result of the expression.

• Solution of a First-Degree Equation: The solution of a first-degree equation is the value that, when substituted for the variable, makes the equality true. For the equation ax + b = c, the solution is x = (c - b)/a.

• Transformation of an Equation: Equations can be manipulated in various ways, changing the position of terms or performing equivalent operations on both sides of the expression. These transformations are useful for isolating the variable x and finding its solution.

Key Terms

• Variable: It is a symbol, usually a letter, that represents an unknown value. In a first-degree equation, this variable is raised to the first power.

• Coefficient: It is the numerical value that multiplies the variable in an expression. In the equation ax + b = c, a is the coefficient of x.

• Independent Term: It is a number, not accompanied by a variable, in an equation or expression. In ax + b = c, b is the independent term.

• Equivalent Operations: These are operations that, although different, have the same solution. When solving an equation, we can perform equivalent operations to isolate the variable.

Examples and Cases

1. Example of a First-Degree Equation: Let's consider the equation 2x + 3 = 7. We can see that the variable x is raised to the first power. The coefficient of x is 2, the independent term is 3, and the result is 7.

2. Solution of a First-Degree Equation: To solve the equation 2x + 3 = 7, first, we subtract 3 from both sides of the equation to isolate the term 2x. We have 2x = 4. Then, by dividing both sides by 2, we get x = 2. Therefore, the solution to this equation is x = 2.

3. Another Equation Transformation: Let's take the equation 4x - 6 = 10. We can move the independent term (-6) to the other side of the equation, changing the sign. Thus, 4x = 16. By dividing both sides by 4, we find that x = 4, the solution of the equation.

Through these examples and the theoretical understanding presented, I hope you can see the beauty and usefulness of first-degree equations, as well as the importance of knowing how to solve them correctly.

Detailed Summary

Relevant Points

• Nature of Linear Equations: The linear equations (ax + b = c) that constitute the theme of this topic are an algebraic expression, where the coefficients (a) and the independent term (b) are multiplied or added with the variable (x), resulting in the value (c). The degree of these equations is one (1st degree) due to the elevation of the variable to the first power.

• Importance of the Coefficient and the Independent Term: The coefficient (a) is crucial in determining the slope of the line associated with the equation on the Cartesian plane, while the independent term (b) defines the point where the line intersects the y-axis, if the equation is plotted.

• Solutions of First-Degree Equations: The solution to a first-degree equation (ax + b = c) is the value of x that makes it true. Isolating the variable x involves performing equivalent operations on both sides of the equation. The formula to find the solution of a linear equation is x = (c - b)/a.

Conclusions

• Mastery of Transferable Skills: Solving first-degree equations problems is not just a mathematical task but also a logical reasoning skill that can be transferred to everyday life. Situations that require the evaluation and solution of linear relationships can range from financial issues (such as calculating discounts) to engineering or physics tasks.

• Importance of Algebraic Manipulation Skills: The ability to manipulate algebraic expressions, such as transforming an equation to isolate the unknown variable, is vital for the successful resolution of first-degree equations. These skills also serve as a basis for more advanced topics in Algebra and Calculus.

Exercises

1. Easy Level Exercise: Solve the equation 3x + 2 = 5. Apply the solution steps described to find the value of x that makes the equation true.

2. Medium Level Exercise: The result of a first-degree equation is 9 when x = 3. What is the value of x if the result of the equation is 20?

3. Hard Level Exercise: The temperature is decreasing at a fixed rate of 2 degrees per hour. The initial temperature was 26 degrees. What temperature is predicted for x hours from now? Model this situation as an equation and solve for x = 4.

Remember, constant practice is essential to improve your skills in solving first-degree equations problems. Feel free to formulate and solve your own problems to consolidate your understanding of this essential topic.