Properties and Procedures Used

Oct 13, 2025 by recoveryshake.com 4 views

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will examine the steps used to solve the equation $12.5x - 10.2 = 3(2.5x + 4.2) - 6$ and analyze the properties or procedures used to complete each step. By breaking down the solution process, we can gain a deeper understanding of the underlying mathematical concepts and improve our problem-solving skills.

Step 1: Distributive Property

The first step in solving the equation is to apply the distributive property to the right-hand side of the equation. The distributive property states that for any real numbers $a$ , $b$ , and $c$ , $a(b + c) = ab + ac$ . In this case, we have:

$12.5x - 10.2 = 3(2.5x + 4.2) - 6$

Using the distributive property, we can rewrite the right-hand side of the equation as:

$12.5x - 10.2 = 7.5x + 12.6 - 6$

Here, we can see that the distributive property was used to expand the product $3(2.5x + 4.2)$ .

Step 2: Combining Like Terms

The next step is to combine like terms on the right-hand side of the equation. Like terms are terms that have the same variable raised to the same power. In this case, we have:

$12.5x - 10.2 = 7.5x + 12.6 - 6$

We can combine the constant terms $12.6$ and $-6$ to get:

$12.5x - 10.2 = 7.5x + 6.6$

Here, we can see that the like terms were combined using the commutative and associative properties of addition.

Step 3: Isolating the Variable

The final step is to isolate the variable $x$ on one side of the equation. To do this, we can add $10.2$ to both sides of the equation:

$12.5x = 7.5x + 6.6 + 10.2$

Simplifying the right-hand side of the equation, we get:

$12.5x = 7.5x + 16.8$

Subtracting $7.5x$ from both sides of the equation, we get:

$5x = 16.8$

Finally, dividing both sides of the equation by $5$ , we get:

$x = 3.36$

Here, we can see that the variable $x$ was isolated using the inverse operations of addition and multiplication.

Conclusion

In this article, we examined the steps used to solve the equation $12.5x - 10.2 = 3(2.5x + 4.2) - 6$ and analyzed the properties or procedures used to complete each step. By breaking down the solution process, we can gain a deeper understanding of the underlying mathematical concepts and improve our problem-solving skills. The distributive property, combining like terms, and isolating the variable are all essential skills for solving linear equations.

Properties and Procedures Used

Distributive Property: The distributive property was used to expand the product $3(2.5x + 4.2)$ .
Commutative and Associative Properties of Addition: The like terms were combined using the commutative and associative properties of addition.
Inverse Operations: The variable $x$ was isolated using the inverse operations of addition and multiplication.

Real-World Applications

Solving linear equations has numerous real-world applications in fields such as physics, engineering, economics, and computer science. For example, in physics, linear equations are used to model the motion of objects, while in engineering, they are used to design and optimize systems. In economics, linear equations are used to model the behavior of markets and economies, and in computer science, they are used to develop algorithms and solve problems.

Tips and Tricks

Use the distributive property to expand products: The distributive property is a powerful tool for expanding products and simplifying expressions.
Combine like terms carefully: Combining like terms requires attention to detail and a clear understanding of the underlying mathematical concepts.
Isolate the variable using inverse operations: Isolating the variable using inverse operations is a crucial step in solving linear equations.

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will provide a Q&A guide to help you understand the steps involved in solving linear equations. Whether you're a student struggling with algebra or a teacher looking for ways to explain complex concepts, this guide is designed to help you navigate the world of linear equations.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable (usually x) is 1. Linear equations can be written in the form ax + b = c, where a, b, and c are constants.

Q: What are the steps to solve a linear equation?

A: The steps to solve a linear equation are:

Distribute: Use the distributive property to expand any products on the right-hand side of the equation.
Combine like terms: Combine any like terms on the right-hand side of the equation.
Isolate the variable: Use inverse operations to isolate the variable on one side of the equation.

Q: What is the distributive property?

A: The distributive property is a mathematical property that allows us to expand a product of a number and a sum. It states that for any real numbers a, b, and c, a(b + c) = ab + ac.

Q: How do I combine like terms?

A: To combine like terms, you need to identify the terms that have the same variable raised to the same power. Then, you can add or subtract the coefficients of those terms.

Q: What are inverse operations?

A: Inverse operations are pairs of operations that "undo" each other. For example, addition and subtraction are inverse operations, as are multiplication and division.

Q: How do I isolate the variable?

A: To isolate the variable, you need to use inverse operations to get the variable on one side of the equation by itself. This may involve adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

Q: What are some common mistakes to avoid when solving linear equations?

A: Some common mistakes to avoid when solving linear equations include:

Not distributing: Failing to distribute the product on the right-hand side of the equation.
Not combining like terms: Failing to combine like terms on the right-hand side of the equation.
Not isolating the variable: Failing to isolate the variable on one side of the equation.

Q: How can I practice solving linear equations?

A: There are many ways to practice solving linear equations, including:

Working through practice problems: Try solving linear equations on your own using practice problems.
Using online resources: There are many online resources available that can help you practice solving linear equations, including video tutorials and interactive exercises.
Seeking help from a teacher or tutor: If you're struggling to understand linear equations, don't be afraid to seek help from a teacher or tutor.

Conclusion

Solving linear equations is a crucial skill for students to master, and with practice and patience, anyone can become proficient. By following the steps outlined in this Q&A guide, you can improve your problem-solving skills and become more confident in your ability to solve linear equations. Remember to always distribute, combine like terms, and isolate the variable to ensure that you're solving the equation correctly.