Solving Equations A Step-by-Step Guide To 3x - 6 = 6x - 9

In the realm of mathematics, solving equations is a fundamental skill. It's like deciphering a code, where we aim to find the value of an unknown variable that makes the equation true. Equations are the bedrock of many mathematical concepts and are essential in various fields, including science, engineering, and economics. This article will delve into the process of solving a specific linear equation: 3x - 6 = 6x - 9. We'll break down each step, providing a clear and concise explanation to help you understand the underlying principles. By mastering these techniques, you'll be well-equipped to tackle more complex equations in the future. The importance of equation-solving extends far beyond the classroom. It's a critical skill for problem-solving in everyday life, from managing personal finances to making informed decisions in various situations. Understanding how to manipulate equations empowers you to analyze situations, identify relationships between variables, and arrive at logical conclusions. This article serves as a comprehensive guide, offering step-by-step instructions, explanations, and insights that will enhance your understanding and proficiency in solving linear equations. Whether you're a student learning the basics or someone looking to refresh your skills, this resource will provide you with the tools and knowledge you need to succeed. Remember, practice is key to mastering equation-solving. Work through the example problems, and don't hesitate to explore other equations to solidify your understanding. With dedication and the right approach, you can confidently tackle any equation that comes your way.

To begin solving the equation 3x - 6 = 6x - 9, our primary goal is to isolate the terms containing the variable 'x' on one side of the equation. This involves strategically moving terms from one side to the other while maintaining the equation's balance. The fundamental principle we rely on is the addition property of equality, which states that adding the same value to both sides of an equation doesn't change its solution. Similarly, the subtraction property of equality allows us to subtract the same value from both sides without affecting the solution. In our equation, we have '3x' on the left side and '6x' on the right side. To consolidate the 'x' terms, we can subtract '3x' from both sides of the equation. This will eliminate the '3x' term on the left side and move all the 'x' terms to the right side. Performing this operation, we get:

3x - 6 - 3x = 6x - 9 - 3x

Simplifying both sides, we have:

-6 = 3x - 9

Now, we have successfully isolated the 'x' term on the right side. The next step involves isolating the constant terms, which are the numbers without any variables attached. In this case, we have '-6' on the left side and '-9' on the right side. To move the '-9' term from the right side to the left side, we'll use the addition property of equality again. We'll add '9' to both sides of the equation. This will cancel out the '-9' on the right side and bring the constant terms to the left side. Adding '9' to both sides, we get:

-6 + 9 = 3x - 9 + 9

Simplifying both sides, we have:

3 = 3x

We have now successfully isolated the 'x' term and the constant term on opposite sides of the equation. The equation is now in a simpler form, making it easier to solve for 'x'. The next step will involve isolating 'x' completely by dividing both sides of the equation by the coefficient of 'x'.

Having consolidated the variable terms and constants, the next crucial step in solving the equation 3 = 3x is to isolate 'x' completely. This means getting 'x' by itself on one side of the equation. The key principle we employ here is the division property of equality. This property states that dividing both sides of an equation by the same non-zero value doesn't alter the equation's solution. In our equation, 'x' is being multiplied by '3'. To undo this multiplication and isolate 'x', we need to divide both sides of the equation by '3'. This will effectively cancel out the '3' on the right side, leaving 'x' alone. Performing this division, we get:

3 / 3 = (3x) / 3

Simplifying both sides, we have:

1 = x

Or, equivalently:

x = 1

We have now successfully isolated 'x' and found its value. This value, x = 1, is the solution to the equation 3x - 6 = 6x - 9. It's the specific value that, when substituted back into the original equation, makes the equation true. This process of isolating the variable is fundamental to solving a wide range of equations. By understanding and applying the division property of equality, you can confidently solve for 'x' in equations where 'x' is being multiplied by a constant. The next step is to verify our solution to ensure that it is correct.

After solving an equation, it's always a good practice to verify the solution. This step ensures that the value we found for 'x' actually satisfies the original equation. It's a crucial step in preventing errors and building confidence in our solution. To verify our solution, x = 1, we substitute this value back into the original equation: 3x - 6 = 6x - 9. Replacing 'x' with '1', we get:

3(1) - 6 = 6(1) - 9

Now, we simplify both sides of the equation separately. On the left side, we have:

3(1) - 6 = 3 - 6 = -3

On the right side, we have:

6(1) - 9 = 6 - 9 = -3

Comparing both sides, we see that:

-3 = -3

Since both sides of the equation are equal when x = 1, we can confidently conclude that our solution is correct. This verification step not only confirms the accuracy of our solution but also reinforces our understanding of the equation and the steps we took to solve it. In more complex equations, verification becomes even more critical. It helps identify any potential errors in our calculations or algebraic manipulations. By consistently verifying your solutions, you'll develop a strong habit of accuracy and attention to detail, which are essential skills in mathematics and beyond. The ability to verify solutions is a testament to a thorough understanding of equation-solving principles and demonstrates a commitment to precision in mathematical work.

In conclusion, we have successfully solved the equation 3x - 6 = 6x - 9, demonstrating a step-by-step approach to solving linear equations. We began by isolating the variable terms, strategically moving terms across the equation while maintaining balance. This involved applying the addition and subtraction properties of equality to consolidate the 'x' terms on one side and the constant terms on the other. Next, we isolated the variable 'x' completely by utilizing the division property of equality. This allowed us to determine the value of 'x' that satisfies the equation. Finally, we verified our solution by substituting the value back into the original equation. This crucial step confirmed the accuracy of our solution and reinforced our understanding of the equation. The solution we found is x = 1. This value, when substituted into the original equation, makes both sides of the equation equal, thus validating our solution. The process of solving equations is a cornerstone of mathematics and is applicable in numerous real-world scenarios. Mastering these techniques empowers you to tackle a wide range of problems, from simple algebraic expressions to complex mathematical models. The key to success in equation-solving lies in understanding the fundamental principles and practicing consistently. By applying these principles and techniques, you can confidently approach any linear equation and find its solution. Remember, equation-solving is not just about finding the answer; it's about developing a logical and systematic approach to problem-solving. This skill is invaluable in various aspects of life, from academic pursuits to professional endeavors. Continue to practice and refine your equation-solving skills, and you'll find yourself well-equipped to tackle any mathematical challenge that comes your way. This article has provided a comprehensive guide to solving linear equations, but the journey of mathematical learning is ongoing. Explore different types of equations, delve deeper into algebraic concepts, and continue to challenge yourself. With dedication and perseverance, you can unlock the power of mathematics and apply it to solve problems in diverse fields.