Solving Linear Equations -9x + 1 = -x + 17 A Step-by-Step Guide

At the heart of algebra lies the fundamental skill of solving equations. Solving equations is a crucial skill in mathematics, and mastering it opens doors to more advanced concepts. This article provides a comprehensive guide on how to solve the linear equation $-9x + 1 = -x + 17$. We will break down each step, making it easy to understand even for those new to algebra. Linear equations, characterized by a variable raised to the first power, form the bedrock of algebraic problem-solving. Among these equations, a common type encountered is one where the variable, typically denoted as 'x,' appears on both sides of the equation. In this guide, we will dissect the step-by-step process of solving the equation $-9x + 1 = -x + 17$, ensuring clarity and comprehension for learners of all levels. Solving equations is not just a mathematical exercise; it's a powerful tool applicable in various real-world scenarios, from calculating finances to understanding scientific phenomena. Equations are a fundamental tool in mathematics and various other fields, representing relationships between variables and constants. This article delves into a detailed, step-by-step solution of the linear equation -9x + 1 = -x + 17, a typical example where the variable 'x' appears on both sides of the equation. Our primary goal is to isolate the variable 'x' on one side of the equation to determine its value. This process involves a series of algebraic manipulations, each aimed at simplifying the equation while maintaining its balance. We will meticulously explain each step, providing clear reasoning and mathematical justifications to ensure a thorough understanding. By mastering these techniques, you'll gain the confidence to tackle a wide range of algebraic problems. Whether you're a student learning algebra for the first time or someone looking to refresh your skills, this guide offers a clear and concise approach to solving linear equations. Let's embark on this algebraic journey together, demystifying the process and empowering you with the knowledge to solve similar problems with ease. The beauty of mathematics lies in its logical structure and the power it gives us to solve complex problems. By understanding the underlying principles of equation solving, we gain a valuable tool that extends far beyond the classroom, influencing fields like science, engineering, and economics. So, let's dive into the world of algebra and unlock the secrets of solving equations!

The initial step in solving the equation $-9x + 1 = -x + 17$ involves gathering all terms containing the variable x on one side and constant terms on the other. Combining like terms is a fundamental principle in algebra. In this case, we aim to consolidate the 'x' terms on one side of the equation and the constant terms on the other. To accomplish this, we add x to both sides of the equation. This action effectively eliminates the -x term on the right side and moves it to the left side, where it can be combined with the existing -9x term. This step is crucial because it simplifies the equation by grouping similar terms together, making it easier to isolate the variable. Adding x to both sides maintains the equation's balance, ensuring that we are not altering the fundamental relationship between the two expressions. After this step, the equation will be in a more manageable form, paving the way for the subsequent steps in solving for x. Let's break down the mechanics of this operation. We start with the equation $-9x + 1 = -x + 17$. By adding x to both sides, we get: $-9x + x + 1 = -x + x + 17$. Simplifying this, we have $-8x + 1 = 17$. Notice how the -x term has disappeared from the right side, and the -9x term has combined with the added x on the left side, resulting in -8x. This process of combining like terms is not just a mechanical step; it's a strategic move that brings us closer to isolating the variable. By grouping similar terms, we reduce the complexity of the equation, making it easier to identify the next steps required to solve for x. In essence, we are transforming the equation into a more simplified and manageable form without changing its underlying meaning or solution. This principle of maintaining balance while manipulating the equation is a cornerstone of algebraic problem-solving. It ensures that each step we take is mathematically sound and leads us closer to the correct answer. As we progress through the steps, you'll see how this initial combination of like terms sets the stage for the remaining operations, ultimately leading us to the solution for x. So, understanding the importance and mechanics of this step is crucial for mastering the art of equation solving.

Following the combination of x terms, the next crucial step is to isolate the variable term. Our goal now is to isolate the term containing x on one side of the equation. In our case, this means isolating the term -8x. Currently, the term -8x is accompanied by the constant +1. To isolate -8x, we need to eliminate this constant term. This is achieved by subtracting 1 from both sides of the equation. Subtracting 1 from both sides maintains the equation's balance, a fundamental principle in algebraic manipulations. This action effectively cancels out the +1 on the left side, leaving the -8x term isolated. This isolation is a pivotal moment in the solving process, as it brings us one step closer to determining the value of x. Once the variable term is isolated, we can then proceed to divide both sides by the coefficient of x, which will ultimately reveal the solution. The importance of isolating the variable term cannot be overstated. It's a strategic maneuver that simplifies the equation and makes it solvable. Think of it as peeling away the layers surrounding the variable, gradually revealing its true value. Let's perform the subtraction: Starting with the equation $-8x + 1 = 17$, we subtract 1 from both sides: $-8x + 1 - 1 = 17 - 1$. This simplifies to $-8x = 16$. Now, we have successfully isolated the -8x term on the left side. The equation is now in a much cleaner and more manageable form. We have effectively separated the variable term from the constant terms, paving the way for the final step of solving for x. This step highlights the power of algebraic manipulation in simplifying complex equations. By applying the principle of maintaining balance and strategically eliminating terms, we can transform an equation into a form that readily reveals the solution. In the broader context of algebraic problem-solving, the ability to isolate variable terms is a key skill. It allows us to tackle a wide range of equations and mathematical challenges. So, understanding the rationale and mechanics of this step is crucial for building a strong foundation in algebra. As we move on to the next step, you'll see how the isolation of the variable term directly leads us to the final solution.

After isolating the variable term, the final step in solving the equation $-9x + 1 = -x + 17$ is to solve for x. Now that we have the equation $-8x = 16$, we need to isolate x completely. This involves dividing both sides of the equation by the coefficient of x, which in this case is -8. Dividing both sides by -8 is a crucial step because it directly reveals the value of x. It's the final algebraic manipulation that separates x from its coefficient, providing us with the solution. This division maintains the equation's balance, ensuring that the value of x we obtain is accurate. By performing this step, we are essentially undoing the multiplication that was previously applied to x, thus revealing its isolated value. The significance of this step lies in its ability to provide a definitive answer to the equation. It's the culmination of all the previous steps, where we strategically simplified and manipulated the equation to arrive at this point. Let's execute the division: Starting with the equation $-8x = 16$, we divide both sides by -8: $(-8x) / -8 = 16 / -8$. This simplifies to $x = -2$. Therefore, the solution to the equation $-9x + 1 = -x + 17$ is x = -2. This result signifies the value of x that satisfies the original equation. In other words, when x is -2, both sides of the equation are equal. We can verify this by substituting -2 back into the original equation. This process of solving for x demonstrates the power of algebraic manipulation in unraveling the unknowns within an equation. It's a systematic approach that, when followed correctly, leads to a precise and accurate solution. This final step underscores the importance of each preceding step, highlighting how they all contribute to the ultimate goal of finding the value of x. In the broader context of mathematics, solving for variables is a fundamental skill that underpins many advanced concepts. It's a building block for problem-solving in various fields, including science, engineering, and economics. So, mastering this skill is essential for anyone pursuing further studies in mathematics or related disciplines. As you continue your mathematical journey, you'll encounter more complex equations and problems. However, the fundamental principles of equation solving, as demonstrated in this example, will remain invaluable tools in your arsenal.

To ensure the accuracy of our solution, it's essential to verify by substituting the solution back into the original equation. Substituting the solution involves replacing the variable x in the original equation with the value we obtained, which in this case is -2. This process allows us to check if both sides of the equation are equal when the value of x is substituted. If the equation holds true, then our solution is correct. Verification is a crucial step because it helps us catch any errors that may have occurred during the solving process. It provides a final confirmation that our algebraic manipulations were accurate and that the value we found for x is indeed the solution. This step is particularly important when dealing with more complex equations, where the chances of making a mistake are higher. Let's perform the substitution: Starting with the original equation $-9x + 1 = -x + 17$, we substitute x with -2: $-9(-2) + 1 = -(-2) + 17$. Simplifying the left side, we get: $18 + 1 = 19$. Simplifying the right side, we get: $2 + 17 = 19$. Since both sides of the equation are equal (19 = 19), our solution x = -2 is verified. This verification process not only confirms the accuracy of our solution but also reinforces our understanding of the equation and its properties. It demonstrates that the value we found for x is the one that makes the equation true. In the broader context of mathematical problem-solving, verification is a valuable habit to cultivate. It promotes accuracy and attention to detail, which are essential qualities for success in mathematics and other fields. By taking the time to verify our solutions, we can build confidence in our abilities and minimize the risk of errors. As you encounter more challenging problems, the importance of verification will become even more apparent. It's a safeguard against mistakes and a powerful tool for ensuring the correctness of our work. So, make it a practice to always verify your solutions, and you'll be well on your way to mastering the art of mathematical problem-solving.

In conclusion, mastering linear equations is a fundamental skill in algebra and mathematics as a whole. We have successfully solved the equation $-9x + 1 = -x + 17$ by following a series of logical steps: combining like terms, isolating the variable term, solving for x, and verifying the solution. This step-by-step approach provides a clear and structured method for tackling linear equations. By understanding the underlying principles of each step, you can confidently apply these techniques to solve a wide range of algebraic problems. The ability to solve linear equations is not just a mathematical exercise; it's a valuable tool that can be applied in various real-world scenarios. From calculating finances to understanding scientific phenomena, equations play a crucial role in problem-solving. As you continue your mathematical journey, the skills you've learned here will serve as a strong foundation for more advanced concepts. Linear equations are the building blocks of algebra, and mastering them will open doors to more complex topics such as systems of equations, inequalities, and quadratic equations. Remember, practice is key to mastering any mathematical skill. The more you practice solving linear equations, the more confident and proficient you will become. Don't be afraid to tackle challenging problems, and always take the time to verify your solutions. The journey of learning mathematics is a continuous process of building knowledge and skills. By embracing challenges and persevering through difficulties, you can achieve your mathematical goals. So, continue to explore the world of mathematics, and never stop learning! This concludes our comprehensive guide on solving the equation $-9x + 1 = -x + 17$. We hope this article has provided you with a clear understanding of the process and the confidence to tackle similar problems. Keep practicing, and you'll soon master the art of solving linear equations.

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