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

Hey guys! Ever stumbled upon an equation that looks like a jumbled mess of numbers and variables? Don't worry, we've all been there! Today, we're going to break down a common type of equation and solve it step-by-step. Specifically, we'll be tackling the equation $-9x + 1 = -x + 17$. Trust me, it's not as scary as it looks! We'll use some basic algebraic principles to isolate the variable x and find its value. This process involves moving terms around, combining like terms, and ultimately getting x all by itself on one side of the equation. By the end of this article, you'll not only know how to solve this particular equation but also have a solid foundation for tackling similar algebraic problems. So, grab your pencils and let's dive in!

Understanding the Basics of Algebraic Equations

Before we jump into the specifics of solving $-9x + 1 = -x + 17$, let's quickly review some fundamental concepts about algebraic equations. Think of an equation as a balanced scale. The equals sign (=) is the fulcrum, and the expressions on either side are the weights. Our goal is to keep the scale balanced while we manipulate the equation to find the value of the unknown variable, in this case, x. This means that whatever operation we perform on one side of the equation, we must perform the exact same operation on the other side. This principle is the cornerstone of solving algebraic equations. Variables, like x, represent unknown quantities. Coefficients are the numbers that multiply the variables (e.g., -9 in -9x). Constants are the standalone numbers (e.g., 1 and 17 in our equation). Our objective is to isolate the variable on one side of the equation, meaning we want to get x all by itself. To do this, we'll use inverse operations. Addition and subtraction are inverse operations, as are multiplication and division. By applying these inverse operations strategically, we can "undo" the operations that are affecting x and gradually isolate it. For instance, if we have a term being added to x, we can subtract that term from both sides of the equation to eliminate it. Similarly, if x is being multiplied by a number, we can divide both sides by that number. Remember, the key is to maintain balance by performing the same operation on both sides of the equation. This ensures that the equation remains true and that we arrive at the correct solution for x.

Step-by-Step Solution: $-9x + 1 = -x + 17$

Okay, let's get down to business and solve the equation $-9x + 1 = -x + 17$ step-by-step. We'll break it down into manageable chunks so you can follow along easily. Our primary goal here is to isolate x on one side of the equation. The first thing we want to do is gather all the x terms on one side and all the constant terms on the other. A common strategy is to move the x terms to the side that has the larger coefficient (in absolute value) to avoid dealing with negative coefficients later on. In this case, -9 is less than -1, so let's move the $-x$ term from the right side to the left side. To do this, we'll add x to both sides of the equation. Remember, whatever we do to one side, we must do to the other to maintain balance! This gives us: $-9x + 1 + x = -x + 17 + x$, which simplifies to $-8x + 1 = 17$. Now, we need to isolate the x term further. We have a constant term, +1, on the same side as the x term. To get rid of it, we'll subtract 1 from both sides of the equation: $-8x + 1 - 1 = 17 - 1$. This simplifies to $-8x = 16$. We're almost there! Now, x is being multiplied by -8. To isolate x, we need to perform the inverse operation, which is division. We'll divide both sides of the equation by -8: $(-8x) / -8 = 16 / -8$. This simplifies to $x = -2$. And there you have it! We've successfully solved the equation. The value of x that makes the equation true is -2. To be absolutely sure of our answer, it's always a good idea to check our solution by plugging it back into the original equation. We'll do that in the next section.

Verifying the Solution

Alright, we've arrived at the solution $x = -2$, but how do we know for sure that it's correct? The best way to confirm our answer is to plug it back into the original equation and see if it holds true. This process is called verifying the solution, and it's a crucial step in solving any algebraic equation. It's like double-checking your work to make sure you didn't make any mistakes along the way. So, let's take our original equation, $-9x + 1 = -x + 17$, and substitute $-2$ for x. This gives us: $-9(-2) + 1 = -(-2) + 17$. Now, we'll simplify both sides of the equation separately. On the left side, we have $-9(-2)$, which equals 18. So, the left side becomes $18 + 1$, which equals 19. On the right side, we have $-(-2)$, which equals 2. So, the right side becomes $2 + 17$, which also equals 19. Therefore, our equation now reads $19 = 19$. Bingo! The left side is equal to the right side, which means our solution, $x = -2$, is indeed correct. This verification step gives us confidence that we've solved the equation accurately. If the two sides of the equation didn't match after substituting our solution, it would indicate that we made an error somewhere in our steps, and we'd need to go back and re-examine our work. So, remember to always verify your solutions, especially in more complex equations, to ensure you've got the right answer. It's a simple yet powerful technique that can save you from making costly mistakes.

Common Mistakes to Avoid

Solving equations can be tricky, and it's easy to make mistakes if you're not careful. To help you avoid common pitfalls, let's discuss some frequent errors that students make when tackling algebraic equations. One of the most common mistakes is forgetting to apply an operation to both sides of the equation. Remember, the equals sign represents a balance, so whatever you do to one side, you must do to the other. For instance, if you subtract a number from the left side, you need to subtract the same number from the right side. Failing to do so will disrupt the balance and lead to an incorrect solution. Another frequent error is incorrectly combining like terms. Like terms are terms that have the same variable raised to the same power (e.g., -9x and -x are like terms, but -9x and 1 are not). When combining like terms, make sure you only add or subtract the coefficients. For example, $-9x + x$ simplifies to $-8x$, not $-10x$. Sign errors are also a common source of mistakes. Be especially careful when dealing with negative numbers. Remember the rules for multiplying and dividing negative numbers: a negative times a negative is a positive, and a negative times a positive is a negative. Similarly, a negative divided by a negative is a positive, and a negative divided by a positive is a negative. Another mistake is distributing incorrectly. If you have a number multiplying a group of terms inside parentheses, you need to multiply that number by each term inside the parentheses. For example, if you have 2(x + 3), you need to distribute the 2 to both the x and the 3, resulting in 2x + 6. Finally, don't forget to verify your solution! Plugging your answer back into the original equation is a simple yet effective way to catch any errors you might have made along the way. By being aware of these common mistakes and taking extra care with each step, you can significantly improve your accuracy in solving algebraic equations.

Practice Problems and Further Learning

Now that we've conquered the equation $-9x + 1 = -x + 17$, it's time to put your newfound skills to the test! Practice makes perfect, so the more you work through similar problems, the more confident and proficient you'll become. To get you started, here are a few practice problems that are similar in structure to the one we just solved:

1. Solve for x: $-5x + 3 = -2x + 9$
2. Solve for y: $4y - 7 = -y + 13$
3. Solve for z: $-2z + 6 = z - 3$

Remember to follow the same steps we outlined earlier: gather the variable terms on one side, gather the constant terms on the other side, and then isolate the variable by performing the appropriate inverse operations. Don't forget to verify your solutions by plugging them back into the original equations! If you're looking for more resources to further your understanding of algebra, there are plenty of options available. Many websites offer free math tutorials and practice problems. Khan Academy, for example, has a comprehensive library of videos and exercises covering a wide range of math topics, including algebra. You can also find helpful textbooks and workbooks at your local library or bookstore. If you're struggling with a particular concept, consider seeking help from a math tutor or your teacher. They can provide personalized guidance and support to help you overcome your challenges. Online forums and communities dedicated to mathematics can also be valuable resources. You can ask questions, share your solutions, and learn from others. Remember, learning algebra is a journey, and it takes time and effort. Don't get discouraged if you don't understand everything right away. Keep practicing, keep asking questions, and you'll gradually build your skills and confidence.

By working through these practice problems and exploring additional resources, you'll solidify your understanding of how to solve linear equations and pave the way for success in more advanced algebraic concepts. Keep up the great work, and happy solving!