Solving 6(x-1) = 9(x+2) A Step-by-Step Guide

JU07/09/2025 08, 2025 by THE IDEN 45 views

Solving Linear Equations A Step-by-Step Guide to 6(x-1) = 9(x+2)

Introduction

In the realm of mathematics, solving equations is a fundamental skill. Equations are the backbone of various mathematical concepts, and the ability to solve them opens doors to more advanced topics. This article dives deep into the process of solving a specific linear equation: 6(x-1) = 9(x+2). We will break down each step in detail, ensuring a clear and comprehensive understanding for readers of all levels. Our primary focus is not just to provide the solution, but to equip you with the knowledge and confidence to tackle similar equations independently. Understanding the underlying principles and techniques is crucial for success in mathematics, as it allows you to approach problems with a logical and systematic mindset. This guide will serve as a valuable resource for students, educators, and anyone seeking to enhance their problem-solving skills. We will explore the concepts of distribution, combining like terms, and isolating the variable, which are essential components of solving linear equations. The goal is to make the process accessible and engaging, fostering a deeper appreciation for the elegance and power of algebraic manipulation. So, let's embark on this mathematical journey and unravel the solution to 6(x-1) = 9(x+2). Remember, practice is key, and the more you engage with these concepts, the more proficient you will become. Our approach will emphasize clarity and precision, ensuring that each step is well-explained and easy to follow. By mastering these fundamental skills, you will be well-prepared to tackle more complex mathematical challenges in the future. The importance of understanding the step-by-step process cannot be overstated, as it forms the foundation for more advanced mathematical concepts. We will also highlight common pitfalls and mistakes that students often make, providing valuable insights into how to avoid them. So, let’s get started and unlock the secrets to solving linear equations!

Understanding the Equation

The equation we aim to solve is 6(x-1) = 9(x+2). This is a linear equation in one variable, x. Linear equations are characterized by the fact that the highest power of the variable is 1. This particular equation involves parentheses, which means we'll need to use the distributive property to simplify it. The distributive property states that for any numbers a, b, and c, a(b + c) = ab + ac. Applying this property is crucial for removing the parentheses and making the equation easier to work with. Before we jump into the solution, let's take a moment to understand the structure of the equation. On the left-hand side, we have 6 multiplied by the expression (x-1). On the right-hand side, we have 9 multiplied by the expression (x+2). Our goal is to isolate the variable x on one side of the equation. This means we need to perform a series of algebraic operations to get x by itself. These operations must be done in a way that maintains the equality of the equation, meaning whatever we do to one side, we must also do to the other side. Understanding the balance in the equation is paramount. Think of it as a seesaw; if you add weight to one side, you must add the same weight to the other side to keep it balanced. This principle is fundamental to solving any algebraic equation. The equation 6(x-1) = 9(x+2) represents a mathematical relationship between the expressions on either side of the equals sign. Our task is to find the value of x that makes this relationship true. This value is the solution to the equation. By systematically applying algebraic principles, we can unravel this relationship and determine the value of x. It's important to visualize the equation as a statement that needs to be made true. Our algebraic manipulations are the tools we use to uncover the truth. The beauty of algebra lies in its ability to transform equations into simpler forms while preserving their underlying meaning. This allows us to gradually isolate the variable and reveal its value. So, with a clear understanding of the equation's structure and the principles of equality, we are well-prepared to proceed with the step-by-step solution.

Step-by-Step Solution

Apply the Distributive Property: The first step in solving this equation is to eliminate the parentheses. We do this by applying the distributive property. This involves multiplying the number outside the parentheses by each term inside the parentheses. For the left side, we multiply 6 by both x and -1. This gives us 6 * x - 6 * 1, which simplifies to 6x - 6. For the right side, we multiply 9 by both x and 2. This gives us 9 * x + 9 * 2, which simplifies to 9x + 18. Now, our equation looks like this: 6x - 6 = 9x + 18. The distributive property is a fundamental tool in algebra, allowing us to expand expressions and simplify equations. It's crucial to apply this property correctly, ensuring that each term inside the parentheses is multiplied by the factor outside. A common mistake is to forget to multiply all the terms, which can lead to an incorrect solution. The application of the distributive property is not just a mechanical step; it's a way of rewriting the equation in a more manageable form. By removing the parentheses, we can now proceed with combining like terms and isolating the variable. This step highlights the power of algebraic manipulation in transforming complex expressions into simpler ones. Remember, the goal is to maintain the equality of the equation throughout the process. Whatever operation we perform on one side, we must also perform on the other side. This ensures that the solution we find is valid. The distributive property is a key building block in solving a wide range of algebraic equations. Mastering this property is essential for success in algebra and beyond. So, let's move on to the next step, where we will continue to simplify the equation.
Move Variables to One Side: Our next goal is to gather all the x terms on one side of the equation and all the constant terms on the other side. To do this, we can subtract 6x from both sides of the equation. This will eliminate the x term on the left side. Subtracting 6x from both sides of 6x - 6 = 9x + 18 gives us: 6x - 6 - 6x = 9x + 18 - 6x. This simplifies to -6 = 3x + 18. Now, we have all the x terms on the right side of the equation. The strategy of moving variables to one side is a crucial technique in solving linear equations. It allows us to isolate the variable and eventually determine its value. This step involves applying the principle of equality, ensuring that whatever operation we perform on one side, we also perform on the other side. Subtracting 6x from both sides was a deliberate choice, aiming to eliminate the x term on the left side. We could have also subtracted 9x from both sides, but this would have resulted in a negative coefficient for the x term, which could make the subsequent steps slightly more complicated. Choosing the most efficient path is a hallmark of good problem-solving skills in mathematics. The goal is to simplify the equation as much as possible, making it easier to isolate the variable. This step also highlights the importance of maintaining the balance of the equation. Each operation we perform must be done equally on both sides to preserve the equality. The concept of moving variables to one side is a fundamental technique that is used extensively in algebra. Mastering this technique is essential for solving a wide range of equations. So, let's proceed to the next step, where we will isolate the constant terms on the other side of the equation.
Isolate the Variable Term: Now that we have the equation -6 = 3x + 18, we need to isolate the term with the variable, which is 3x. To do this, we subtract 18 from both sides of the equation. This will eliminate the constant term on the right side. Subtracting 18 from both sides of -6 = 3x + 18 gives us: -6 - 18 = 3x + 18 - 18. This simplifies to -24 = 3x. We have now successfully isolated the variable term on the right side of the equation. The process of isolating the variable term is a critical step in solving linear equations. It involves performing operations on both sides of the equation to eliminate any constant terms that are on the same side as the variable term. Subtracting 18 from both sides was the key to achieving this isolation in our equation. This step reinforces the principle of equality, ensuring that the equation remains balanced throughout the process. Each operation we perform brings us closer to the solution, gradually isolating the variable and revealing its value. The ability to strategically isolate the variable term is a hallmark of proficient algebraic manipulation. It requires a clear understanding of the equation's structure and the operations that will effectively simplify it. This step also highlights the importance of careful arithmetic. Accurate calculations are essential for obtaining the correct solution. A small error in arithmetic can lead to a completely different answer. So, double-checking our calculations is always a good practice. The technique of isolating the variable term is a fundamental tool in algebra and is used extensively in solving various types of equations. Mastering this technique is crucial for success in algebra and beyond. With the variable term now isolated, we are just one step away from finding the solution. Let's move on to the final step, where we will solve for x.
Solve for x: The final step is to solve for x. We have the equation -24 = 3x. To isolate x, we need to divide both sides of the equation by 3. Dividing both sides of -24 = 3x by 3 gives us: -24 / 3 = 3x / 3. This simplifies to -8 = x. Therefore, the solution to the equation is x = -8. We have successfully solved for x by performing the necessary algebraic operations. This final step demonstrates the culmination of our efforts to isolate the variable and determine its value. Dividing both sides by 3 was the key to unlocking the solution. This step reinforces the principle of equality, ensuring that the equation remains balanced throughout the process. The solution x = -8 represents the value that makes the original equation true. This means that if we substitute -8 for x in the equation 6(x-1) = 9(x+2), both sides of the equation will be equal. The ability to confidently solve for x is a fundamental skill in algebra. It requires a clear understanding of the principles of equality and the operations that can be used to isolate the variable. This step also highlights the importance of accuracy. Double-checking our calculations is essential to ensure that we have obtained the correct solution. The solution x = -8 is not just a number; it's the answer to a mathematical puzzle. It's the value that satisfies the equation and makes the mathematical statement true. With this solution in hand, we have completed the process of solving the linear equation. Let's now move on to verifying our solution to ensure its correctness. So, we have successfully navigated the steps to solve for x. The result, x = -8, is the value that satisfies the original equation. But before we declare victory, it's crucial to verify our solution.

Verification

To verify our solution, we substitute x = -8 back into the original equation, 6(x-1) = 9(x+2). Substituting x = -8 into the equation gives us: 6(-8 - 1) = 9(-8 + 2). Now, we simplify both sides of the equation. On the left side, we have 6(-9), which equals -54. On the right side, we have 9(-6), which also equals -54. Since both sides of the equation are equal (-54 = -54), our solution x = -8 is correct. Verification is a crucial step in the problem-solving process. It allows us to confirm that our solution is valid and that we have not made any errors in our calculations. Substituting the solution back into the original equation and checking if both sides are equal is the most reliable way to verify a solution. This step highlights the importance of accuracy and attention to detail in mathematics. A small error in our calculations can lead to an incorrect solution, and verification helps us catch these errors before they become a problem. The process of verification also reinforces our understanding of the equation and its solution. It helps us see that the solution is not just a number, but a value that makes the equation true. This step also emphasizes the importance of showing our work. By writing down each step of the solution, we can easily trace back our steps and identify any errors if our verification fails. Verification is not just a formality; it's an integral part of the problem-solving process. It gives us confidence in our solution and ensures that we have answered the question correctly. So, with our solution verified, we can confidently conclude that x = -8 is the correct answer to the equation 6(x-1) = 9(x+2). The verification process not only confirms the correctness of the solution but also provides a deeper understanding of the equation itself. It solidifies the connection between the solution and the original problem. So, now that we have both solved and verified the equation, we can be assured of our answer.

Conclusion

In this article, we have walked through the process of solving the linear equation 6(x-1) = 9(x+2). We started by understanding the equation, then applied the distributive property, moved variables to one side, isolated the variable term, solved for x, and finally, verified our solution. The solution we found is x = -8. Solving linear equations is a fundamental skill in mathematics, and this article has provided a comprehensive guide to the process. We have broken down each step in detail, explaining the underlying principles and techniques. This approach not only helps you solve this specific equation but also equips you with the knowledge and confidence to tackle similar equations in the future. The key takeaways from this article include the importance of the distributive property, the principle of equality, the strategy of isolating the variable, and the crucial step of verification. These concepts are essential for success in algebra and beyond. Remember, practice is key. The more you engage with these concepts, the more proficient you will become. Solving equations is like learning a new language; it takes time and effort, but the rewards are well worth it. The ability to solve equations confidently opens doors to more advanced mathematical concepts and applications. Mathematics is a journey, not a destination. Each equation we solve is a step forward on that journey. This article has aimed to provide you with a clear and engaging guide to solving linear equations, and we hope that it has been helpful. So, keep practicing, keep exploring, and keep solving!