Solving 5x + 8 - 3x = -10 A Step By Step Guide

Hey guys! Ever find yourself staring at an equation and feeling totally lost? Don't worry, we've all been there! Today, we're going to break down a common type of equation and show you step-by-step how to solve it. We're talking about the equation 5x + 8 - 3x = -10. Sounds intimidating? Trust me, it's not! By the end of this guide, you'll be solving equations like a pro. We'll cover everything from the basic principles to the nitty-gritty details, ensuring you understand not just how to solve it, but why each step works. So, grab a pen and paper, and let's dive in! We're about to unravel the mystery behind this equation and equip you with the skills to tackle similar problems with confidence. The journey of understanding algebra can be exciting, and this equation is a fantastic starting point. Remember, practice makes perfect, so the more equations you solve, the easier it will become. Let's get started and transform those equation-solving woes into wins! We're not just solving for 'x' here; we're building a foundation for more advanced mathematical concepts. Think of it as unlocking a new level in your mathematical journey. Each step we take is a building block, strengthening your understanding and problem-solving abilities. This isn't just about getting the right answer; it's about developing a logical approach that you can apply to any equation that comes your way. So, let's approach this with a spirit of curiosity and a willingness to learn. We'll break down each step, explain the reasoning behind it, and make sure you're comfortable before moving on. By the end of this, you'll not only be able to solve this specific equation, but you'll also have a solid framework for tackling algebraic problems in general.

Understanding the Basics: What is an Equation?

Before we jump into solving 5x + 8 - 3x = -10, let's make sure we're all on the same page about what an equation actually is. Think of an equation like a balanced scale. On one side, you have an expression (like 5x + 8 - 3x), and on the other side, you have another expression (in this case, -10). The equals sign (=) is the fulcrum, the point of balance. The whole goal of solving an equation is to figure out what value of x will keep the scale perfectly balanced. In other words, we want to find the value of x that makes the left side of the equation equal to the right side. Equations are the fundamental building blocks of algebra, and understanding them is key to unlocking more advanced mathematical concepts. They represent relationships between quantities and allow us to solve for unknown values. When you see an equation, think of it as a puzzle waiting to be solved. Each step we take is a move towards finding the solution, the value of the unknown variable that makes the equation true. And just like a puzzle, there's often a logical sequence of steps that will lead us to the answer. The beauty of equations is that they provide a framework for solving problems in a structured way. We can manipulate the equation using algebraic rules, keeping the balance intact, until we isolate the variable we're trying to solve for. This process of isolating the variable is at the heart of equation solving, and it's a skill that will serve you well in many areas of mathematics and beyond. So, as we delve into solving our equation, remember that we're not just finding a number; we're uncovering a fundamental truth about the relationship between the expressions on either side of the equals sign. This understanding will empower you to tackle more complex problems with confidence and clarity.

Step-by-Step Solution: Solving 5x + 8 - 3x = -10

Okay, guys, let's get down to business and solve the equation 5x + 8 - 3x = -10. We'll break it down into easy-to-follow steps. Remember, the key is to keep the equation balanced while we isolate x. Let's start with the first step: Combining Like Terms. This is where we simplify each side of the equation by grouping together terms that have the same variable (in this case, x) and the constant terms (the numbers without variables). On the left side of the equation, we have 5x and -3x. These are like terms because they both have x. When we combine them, we get 5x - 3x = 2x. So, our equation now looks like 2x + 8 = -10. See how much simpler that is already? This step is crucial because it reduces the complexity of the equation and makes it easier to work with. By combining like terms, we're essentially tidying up the equation, making it more manageable. It's like organizing your workspace before starting a project; it helps you focus on the task at hand. Now, let's move on to the next step: Isolating the Variable Term. Our goal is to get the term with x (which is 2x) all by itself on one side of the equation. To do this, we need to get rid of the + 8 that's hanging out on the left side. Remember the balance scale? To keep things balanced, we need to do the same thing to both sides of the equation. So, we'll subtract 8 from both sides: 2x + 8 - 8 = -10 - 8. This simplifies to 2x = -18. We're getting closer! This step is all about strategic manipulation. We're using the properties of equality to our advantage, ensuring that the equation remains balanced while we move closer to isolating x. It's like a carefully planned maneuver in a game of chess, where each move brings us closer to checkmate. And now, for the final step: Solving for x. We have 2x = -18. This means "2 times x equals -18". To find x, we need to undo the multiplication. We do this by dividing both sides of the equation by 2: 2x / 2 = -18 / 2. This gives us x = -9. Boom! We solved it! We found the value of x that makes the equation true. This final step is the culmination of all our efforts. It's the moment where the puzzle pieces click into place and the solution is revealed. And the best part is, we can always check our answer by plugging it back into the original equation to make sure it works.

Checking Your Answer: Making Sure x = -9 is Correct

Alright, we've arrived at the solution x = -9, but how can we be absolutely sure we didn't make a mistake along the way? This is where checking your answer comes in super handy! Think of it as the ultimate safety net. It's a way to verify that our solution is correct and that we've maintained the balance of the equation throughout the solving process. So, how do we check our answer? It's simple! We substitute the value we found for x (which is -9) back into the original equation: 5x + 8 - 3x = -10. Let's plug in -9 for x: 5(-9) + 8 - 3(-9) = -10. Now, we just need to simplify the left side of the equation and see if it equals -10. First, let's do the multiplications: 5(-9) = -45 and -3(-9) = 27. So, our equation now looks like: -45 + 8 + 27 = -10. Next, let's add the numbers: -45 + 8 = -37 and -37 + 27 = -10. So, the left side of the equation simplifies to -10, which is exactly what we have on the right side! This means our solution, x = -9, is correct! We've successfully solved the equation and verified our answer. This process of checking your answer is not just a formality; it's a crucial step in the problem-solving process. It gives you confidence in your solution and helps you identify any errors you might have made. By plugging the solution back into the original equation, you're essentially running a test to see if the equation holds true. If it does, you know you're on the right track. If it doesn't, it's a sign that you need to go back and review your steps to find the mistake. So, always remember to check your answer! It's a small investment of time that can save you from making costly errors and ensure that you're mastering the concepts.

Key Takeaways: Mastering Equation Solving

Okay, guys, we've successfully navigated the equation 5x + 8 - 3x = -10! But the real goal here is to equip you with the skills to tackle any equation that comes your way. So, let's recap the key takeaways and solidify your understanding of equation solving. First and foremost, remember the importance of balancing. An equation is like a scale, and the equals sign is the fulcrum. Whatever operation you perform on one side, you must perform on the other side to maintain the balance. This is the golden rule of equation solving, and it's essential for arriving at the correct solution. Think of it as the foundation upon which all equation-solving techniques are built. Next, combining like terms is your secret weapon for simplifying equations. By grouping together terms with the same variable and constant terms, you can make the equation much more manageable. This step is like decluttering your equation, making it easier to see the path to the solution. It's a simple yet powerful technique that can transform a complex equation into a more approachable one. Then, there's the art of isolating the variable. This is where we strategically manipulate the equation to get the term with the variable all by itself on one side. This often involves using inverse operations (like adding or subtracting to undo addition or subtraction, and multiplying or dividing to undo multiplication or division). This step is like peeling away the layers of an onion, revealing the core variable that we're trying to solve for. It requires a bit of algebraic finesse, but with practice, it becomes second nature. And finally, never, ever forget to check your answer! This is your ultimate safety net, the final step that confirms your solution and catches any potential errors. By substituting your solution back into the original equation, you can verify that it makes the equation true. This step is like the quality control check in a manufacturing process, ensuring that the final product meets the required standards. It's a simple yet crucial step that can save you from making costly mistakes. By mastering these key takeaways, you'll not only be able to solve equations like a pro, but you'll also develop a deeper understanding of algebraic principles. This knowledge will serve you well as you continue your mathematical journey and tackle more complex problems.

Practice Makes Perfect: More Equations to Try

Alright, awesome job making it this far, guys! You've conquered the equation 5x + 8 - 3x = -10, and you've got the key takeaways under your belt. But remember, the best way to truly master equation solving is through practice, practice, practice! Think of it like learning a new instrument or sport; the more you practice, the better you become. So, let's put your newfound skills to the test with some more equations to try. Don't worry, we're not going to throw you into the deep end just yet. We'll start with some equations that are similar to the one we just solved, and then we'll gradually increase the complexity. Grab a pen and paper, and let's dive in! Here are a few equations to get you started: 1. 2x + 5 - x = 12 2. 7y - 3 + 2y = -21 3. 4a + 9 - a = 0 4. 6b - 2 - 4b = 8 Remember to follow the same steps we used earlier: Combine like terms, isolate the variable term, solve for the variable, and most importantly, check your answer! As you work through these equations, pay attention to the patterns and strategies that emerge. Notice how the same principles apply to different equations, even if they look slightly different. This is the beauty of algebra; once you understand the underlying concepts, you can apply them to a wide range of problems. And don't be afraid to make mistakes! Mistakes are a natural part of the learning process. When you encounter an error, take it as an opportunity to learn and grow. Go back and review your steps, try to identify where you went wrong, and then try again. The more you practice, the more confident and proficient you'll become in solving equations. So, keep at it, guys! You've got this! And remember, the journey of learning mathematics is not just about finding the right answers; it's about developing problem-solving skills and building a foundation for future success. So, embrace the challenge, enjoy the process, and keep practicing!

Conclusion: You've Got This!

Wow, you guys have done an amazing job! We've taken a deep dive into solving the equation 5x + 8 - 3x = -10, and along the way, you've learned some invaluable skills and strategies for tackling algebraic equations. You now understand the importance of balancing equations, combining like terms, isolating variables, and checking your answers. These are the fundamental building blocks of algebra, and you've mastered them! But more importantly, you've developed a problem-solving mindset, a willingness to persevere, and the confidence to tackle new challenges. These are skills that will serve you well not only in mathematics but in all areas of your life. Remember, mathematics is not just about numbers and formulas; it's about logical thinking, critical analysis, and creative problem-solving. It's about developing the ability to break down complex problems into smaller, more manageable steps, and then systematically working your way towards a solution. And that's exactly what you've done today! So, as you continue your mathematical journey, remember the lessons you've learned here. Remember the importance of practice, the value of checking your work, and the power of perseverance. And most importantly, remember that you've got this! You have the ability to conquer any equation, any problem, any challenge that comes your way. So, go out there and embrace the world of mathematics with confidence and enthusiasm. Keep learning, keep growing, and keep pushing yourself to new heights. The possibilities are endless, and you have the potential to achieve amazing things. So, congratulations on your success, and keep up the fantastic work!