Solving -8x + 1 + 6x = -3x + 6 Step-by-Step

Hey guys! Let's dive into a common type of math problem you'll often encounter: solving for a variable in an equation. Specifically, we're going to tackle the equation $-8x + 1 + 6x = -3x + 6$. This might look a little intimidating at first, but don't worry! We'll break it down step by step, making it super easy to understand. Solving algebraic equations is a fundamental skill in mathematics, and mastering it opens doors to more complex concepts. In this comprehensive guide, we'll not only solve this particular equation but also equip you with the strategies and knowledge to confidently solve similar problems. Whether you're a student brushing up on your algebra skills or just someone curious about math, this guide is for you. So, let's get started and unlock the secrets of solving for variables! Remember, the key to success in math is practice and understanding the underlying principles. With a little effort and the right guidance, you'll be solving equations like a pro in no time! We'll begin by simplifying the equation, combining like terms, and isolating the variable. This process involves applying the fundamental properties of equality, ensuring that we maintain the balance of the equation throughout each step. Understanding these properties is crucial for solving any algebraic equation accurately. We will also discuss common mistakes to avoid and provide tips for checking your solutions to ensure they are correct. This comprehensive approach will not only help you solve this specific equation but also build a solid foundation for future mathematical endeavors. So, let's embark on this mathematical journey together and conquer the world of equations!

Simplifying the Equation: Combining Like Terms

The first step in solving any equation is to simplify it as much as possible. For our equation, $-8x + 1 + 6x = -3x + 6$, this means combining the like terms on each side. Remember, like terms are terms that have the same variable raised to the same power. In this case, we have two terms with 'x' on the left side: $-8x$ and $+6x$. Combining these, we get $(-8 + 6)x = -2x$. So, the left side of the equation simplifies to $-2x + 1$. Now our equation looks like this: $-2x + 1 = -3x + 6$. See? Already less scary! This simplification process is crucial because it reduces the complexity of the equation, making it easier to manipulate and solve. By combining like terms, we consolidate the variables and constants, which helps in isolating the variable we're trying to find. This step is not just about making the equation look simpler; it's about making the mathematical operations more manageable and reducing the chances of errors in subsequent steps. Think of it as decluttering your workspace before starting a project – a clean and organized equation is much easier to work with! Moreover, mastering the art of combining like terms is a fundamental skill in algebra. It's a building block that you'll use repeatedly in more advanced topics. So, by focusing on this step, you're not just solving this particular equation; you're honing a skill that will benefit you in various mathematical contexts. Remember, accuracy is key when combining like terms. Pay close attention to the signs (positive or negative) of the coefficients. A small mistake in this step can lead to an incorrect solution. So, take your time, double-check your work, and ensure you've combined the terms correctly. With practice, you'll become a pro at simplifying equations, setting you up for success in solving them!

Isolating the Variable: Moving Terms Around

Now that we've simplified our equation to $-2x + 1 = -3x + 6$, the next step is to isolate the variable 'x' on one side of the equation. This means we need to move all the terms containing 'x' to one side and all the constant terms to the other side. To do this, we'll use the properties of equality, which state that we can add or subtract the same value from both sides of an equation without changing its balance. Let's start by moving the $-3x$ term from the right side to the left side. To do this, we add $3x$ to both sides of the equation: $-2x + 1 + 3x = -3x + 6 + 3x$. This simplifies to $x + 1 = 6$. Great! We're one step closer. Notice how adding $3x$ to both sides effectively canceled out the $-3x$ on the right side, moving the 'x' term to the left. This is a key technique in solving equations. Next, we need to move the constant term, $+1$, from the left side to the right side. To do this, we subtract 1 from both sides of the equation: $x + 1 - 1 = 6 - 1$. This simplifies to $x = 5$. And there you have it! We've successfully isolated the variable and found that $x = 5$. The process of isolating the variable involves strategically adding or subtracting terms from both sides of the equation to gradually separate the variable from the other terms. It's like playing a puzzle, where you carefully move pieces around to achieve a specific arrangement. Each step you take should bring you closer to the solution, with the ultimate goal of having the variable all by itself on one side of the equation. Remember, it's crucial to perform the same operation on both sides of the equation to maintain balance. This ensures that the equation remains true and that you're not changing the fundamental relationship between the variables and constants. Isolating the variable is a fundamental skill in algebra, and mastering it will significantly improve your ability to solve a wide range of equations. So, practice this technique, and you'll become more confident and efficient in your equation-solving endeavors!

Verifying the Solution: Plugging It Back In

We've found that $x = 5$ is the solution to our equation. But how can we be sure we didn't make a mistake along the way? The best way to check our answer is to verify it. This means plugging the value we found for 'x' back into the original equation and seeing if it holds true. Our original equation was $-8x + 1 + 6x = -3x + 6$. Let's substitute $x = 5$ into this equation: $-8(5) + 1 + 6(5) = -3(5) + 6$. Now, we simplify both sides of the equation. On the left side, we have $-8(5) = -40$ and $6(5) = 30$, so the left side becomes $-40 + 1 + 30$. Simplifying further, $-40 + 1 = -39$, and $-39 + 30 = -9$. So, the left side of the equation equals $-9$. On the right side, we have $-3(5) = -15$, so the right side becomes $-15 + 6$. Simplifying, we get $-9$. Aha! Both sides of the equation equal $-9$. This means that our solution, $x = 5$, is correct! Verifying your solution is a crucial step in solving equations. It's like double-checking your work on a test – it ensures that you haven't made any errors and that your answer is accurate. By plugging the solution back into the original equation, you're essentially testing whether the value you found satisfies the equation's conditions. If the equation holds true after the substitution, you can be confident that your solution is correct. If the two sides of the equation don't match up, it means there's a mistake somewhere in your solution process. In this case, you'll need to go back and review your steps to identify the error and correct it. Verifying your solution not only confirms your answer but also reinforces your understanding of the equation-solving process. It helps you solidify your skills and develop a greater sense of confidence in your mathematical abilities. So, always make it a habit to verify your solutions, and you'll become a more accurate and proficient equation solver!

Common Mistakes to Avoid When Solving Equations

Solving equations can be tricky, and it's easy to make mistakes if you're not careful. Let's talk about some common pitfalls to watch out for so you can avoid them. One frequent mistake is forgetting to distribute a negative sign properly. For example, if you have an expression like $-(x + 2)$, you need to distribute the negative sign to both terms inside the parentheses, making it $-x - 2$. Forgetting to do this can lead to an incorrect solution. Another common mistake is combining unlike terms. Remember, you can only combine terms that have the same variable raised to the same power. So, you can combine $3x$ and $5x$, but you can't combine $3x$ and $5x^2$. Mixing these up will throw off your calculations. A third mistake is not performing the same operation on both sides of the equation. The golden rule of equation solving is that whatever you do to one side, you must do to the other. If you add a number to one side but forget to add it to the other, you'll break the balance of the equation and get the wrong answer. Sign errors are also a common culprit. Pay close attention to positive and negative signs, especially when adding, subtracting, multiplying, or dividing. A small sign error can completely change the outcome of the equation. Finally, always double-check your work! It's easy to make a simple arithmetic mistake, so take a moment to review your steps and make sure everything adds up correctly. This can save you from a lot of frustration and ensure you get the right answer. By being aware of these common mistakes and taking steps to avoid them, you'll become a much more accurate and confident equation solver. Remember, practice makes perfect, so keep working at it, and you'll master the art of solving equations in no time!

Tips and Tricks for Mastering Equation Solving

Want to become a whiz at solving equations? Here are some extra tips and tricks to help you on your journey! First off, practice, practice, practice! The more equations you solve, the better you'll become at recognizing patterns and applying the right techniques. It's like learning any new skill – the more you do it, the more natural it becomes. Next, organize your work. Keep your steps neat and tidy, and write everything down clearly. This will help you avoid mistakes and make it easier to review your work if you need to. Use different colors or symbols to highlight important steps or terms. Another handy trick is to estimate your answer before you start solving. This can give you a sense of what the solution should be and help you spot any major errors along the way. If your final answer is way off from your estimate, it's a sign that you need to double-check your work. Don't be afraid to break down complex equations into smaller, more manageable steps. This can make the problem seem less overwhelming and help you focus on each individual operation. Look for opportunities to simplify the equation before you start solving it. This might involve combining like terms, distributing terms, or canceling out common factors. The simpler the equation, the easier it will be to solve. If you're struggling with a particular type of equation, seek out extra help. There are tons of resources available online, in textbooks, and from teachers or tutors. Don't be afraid to ask for assistance when you need it. Finally, be patient and persistent. Solving equations can be challenging, but it's also a rewarding skill to develop. Don't get discouraged if you make mistakes – just learn from them and keep practicing. With the right approach and a little bit of effort, you can master the art of equation solving and unlock a whole new world of mathematical possibilities!

By following these steps and understanding the underlying principles, you can confidently solve the equation $-8x + 1 + 6x = -3x + 6$ and many other similar equations. Remember, practice makes perfect, so keep working at it! And that’s it, guys! You've now got a solid understanding of how to solve equations like this one. Keep practicing, and you'll become a math master in no time!