Summary of First Degree Equations

First Degree Equations | Traditional Summary

Contextualization

First-degree equations are fundamental mathematical tools that help us find unknown values in various everyday situations. A first-degree equation is an expression in the form ax + b = c, where 'a', 'b', and 'c' are real numbers and 'x' is the variable we want to determine. This type of equation is essential for solving basic problems and is widely used in various fields, such as economics, physics, and engineering.

In daily life, first-degree equations can be applied in practical situations, such as calculating the quantity of ingredients in a recipe, determining the change in a purchase, or even understanding taxi fare pricing. For example, taxi fare charges usually involve a fixed rate plus a variable that depends on the distance traveled. Understanding these equations allows you to interpret and solve problems efficiently, facilitating informed decision-making.

Concept of First-Degree Equations

First-degree equations are mathematical expressions in the form ax + b = c, where 'a', 'b', and 'c' are real numbers and 'x' is the variable we want to find. These equations are called 'first degree' because the variable 'x' is raised to the first power (i.e., x¹). When we solve a first-degree equation, we are looking for the value of 'x' that makes the equation true.

To better understand, consider the equation 2x + 3 = 11. Our goal is to find the value of 'x' that satisfies this equality. These equations are fundamental in algebra and are used to introduce more advanced concepts in mathematics.

First-degree equations are widely used in various areas, such as physics, economics, and engineering, as they provide a simple way to model and solve real problems. They serve as the foundation for understanding more complex equations and systems of equations.

• General form: ax + b = c.

• The variable 'x' is raised to the first power.

• Used in various fields, such as physics, economics, and engineering.

Properties of Equality

The properties of equality are mathematical rules that allow us to manipulate equations in a way that keeps the equality true. They include: addition, subtraction, multiplication, and division. These properties are fundamental for solving first-degree equations.

The addition property states that if we add the same value to both sides of an equation, the equality will be maintained. For example, if we have the equation 2x + 3 = 11 and subtract 3 from both sides, we get 2x = 8. The same logic applies to subtraction, multiplication, and division.

These properties are essential because they allow us to isolate the variable 'x' on one side of the equation, simplifying the solving process. Without these properties, it would be much more complicated to find the value of 'x' that makes the equation true.

• Addition Property: add the same value to both sides.

• Subtraction Property: subtract the same value from both sides.

• Multiplication Property: multiply both sides by the same value.

• Division Property: divide both sides by the same value.

Steps to Solve First-Degree Equations

Solving a first-degree equation involves a series of steps that help to isolate the variable 'x'. The main steps are: isolate the variable 'x', simplify the equation, and solve to find the value of 'x'. Let's detail each of these steps.

First, we isolate the variable 'x'. This usually involves moving terms from one side of the equation to the other, using the properties of equality. For example, in the equation 2x + 3 = 11, we subtract 3 from both sides to obtain 2x = 8. The next step is to simplify the equation, which may involve combining like terms or performing arithmetic operations.

Finally, we solve the equation to find the value of 'x'. In the example 2x = 8, we divide both sides by 2 to obtain x = 4. These steps are applied systematically to solve any first-degree equation.

• Isolate the variable 'x'.

• Simplify the equation.

• Solve the equation to find the value of 'x'.

Practical Examples and Applications

First-degree equations have numerous practical applications that help illustrate their importance. A common example is the fare pricing of taxi rides. The fare for a taxi ride typically includes a fixed charge and a variable that depends on the distance traveled. We can represent this situation with a first-degree equation.

For example, if a taxi charges R$ 5.00 for the base fare (fixed charge) and R$ 2.00 per kilometer traveled, the equation that represents the total cost of a ride for 'x' kilometers is C = 5 + 2x. If we want to know the cost of a 10-kilometer ride, we substitute 'x' with 10 in the equation: C = 5 + 2(10), resulting in C = 25. Therefore, the total cost of the ride is R$ 25.00.

In addition to taxi rides, these equations can be applied in situations such as calculating change in a purchase, determining the quantity of ingredients in a recipe, or analyzing discounts in promotions. Understanding these practical applications helps students see the relevance of first-degree equations in everyday life.

• Taxi fare pricing: C = 5 + 2x.

• Calculating change in a purchase.

• Determining the quantity of ingredients in a recipe.

• Analyzing discounts in promotions.

To Remember

• First-Degree Equation: An expression in the form ax + b = c, where 'x' is the variable.

• Properties of Equality: Mathematical rules that allow manipulating equations while maintaining equality.

• Isolate the Variable: The process of moving terms from one side of the equation to the other to find the value of 'x'.

• Pricing: The method of calculating the cost of a service, often using a first-degree equation.

Conclusion

First-degree equations are fundamental mathematical tools that enable the resolution of basic and practical problems in everyday life. They are expressions in the form ax + b = c, where 'a', 'b', and 'c' are real numbers and 'x' is the variable we want to determine. Understanding these equations is essential to solving problems in various fields, such as economics, physics, and engineering.

During the class, we addressed the properties of equality, which are fundamental for manipulating and solving first-degree equations. We used practical examples, such as taxi fare pricing, to illustrate the application of these equations in real situations. This helps to see the relevance of theoretical content in everyday life.

The knowledge acquired in this lesson is important not only for understanding more advanced mathematical concepts but also for practical application in daily life. Solving first-degree equations helps make informed and precise decisions in everyday situations, such as calculating expenses, planning finances, and understanding service charges.

Study Tips

• Practice solving different types of first-degree equations to gain confidence and fluency in the procedures.

• Review the properties of equality and how to apply them at different stages of solving equations.

• Try to create your own practical situations where first-degree equations can be applied, such as personal budgeting or discount calculations.