Solving -5(1-5x) + 5(-8x-2) = -4x - 8x A Step-by-Step Guide

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Introduction

Hey guys! Let's dive into solving this equation: -5(1-5x) + 5(-8x-2) = -4x - 8x. Equations might seem intimidating at first, but breaking them down step-by-step makes the process super manageable. In this guide, we'll walk through each stage, ensuring you understand the logic and techniques involved. Our main keyword here is understanding how to solve algebraic equations, and we’ll tackle this specific problem to illustrate the general methods. We aim to provide a clear, comprehensive explanation that’s perfect for anyone learning algebra. So, let’s get started and unravel this mathematical puzzle together! Equations are the backbone of algebra, and mastering them opens doors to more advanced topics. Whether you're a student tackling homework or someone brushing up on math skills, this detailed walkthrough is tailored for you. By the end of this article, you'll not only know how to solve this particular equation but also gain valuable insights into solving similar problems. Remember, the key to math is practice, so feel free to try these methods on other equations too! Let’s make math less daunting and more engaging. We'll take it one step at a time, ensuring that every concept is crystal clear. From understanding the distributive property to combining like terms, we've got you covered. And don't worry if you stumble along the way; that's perfectly normal. The goal is to learn and grow, and this guide is designed to help you do just that. So, grab a pen and paper, and let's get to work!

Step 1: Distribute the Numbers

First up, we need to distribute the numbers outside the parentheses. This involves multiplying the number directly outside the parenthesis by each term inside the parenthesis. So, we have two distributions to take care of: -5 multiplied by (1-5x) and 5 multiplied by (-8x-2). Let's break this down further. When we distribute -5 across (1-5x), we get: -5 * 1 = -5 and -5 * -5x = 25x. Remember, a negative times a negative results in a positive, hence the 25x. Next, let's tackle the second distribution: 5 multiplied by (-8x-2). Here, 5 * -8x = -40x and 5 * -2 = -10. Putting it all together, the equation now looks like this: -5 + 25x - 40x - 10 = -4x - 8x. See how we've transformed the equation by simply applying the distributive property? This is a fundamental step in solving many algebraic equations. Understanding how to distribute correctly is crucial because it simplifies the equation, making it easier to solve. Without this step, we'd be stuck with the parentheses, which can be a bit of a headache. The distributive property allows us to remove those parentheses and work with individual terms. So, make sure you're comfortable with this step before moving on. It's like laying the groundwork for a sturdy building; a solid foundation (in this case, distribution) is essential for the rest of the process. Now that we've handled the distribution, the next step involves combining the like terms. This will further simplify our equation and bring us closer to the final solution. Keep up the great work, guys! We're making excellent progress.

Step 2: Combine Like Terms

Now that we've distributed the numbers, it's time to combine like terms. This means grouping together terms that have the same variable (in this case, x) and constant terms (the numbers without any variables). Looking at our equation, -5 + 25x - 40x - 10 = -4x - 8x, we can identify the like terms on each side. On the left side, we have 25x and -40x, which are both x-terms, and -5 and -10, which are constant terms. Combining 25x and -40x gives us -15x. Similarly, combining -5 and -10 gives us -15. So, the left side of the equation simplifies to -15x - 15. On the right side, we have -4x and -8x. Combining these gives us -12x. Thus, the equation now looks much simpler: -15x - 15 = -12x. See how much cleaner and easier to manage the equation is now? Combining like terms is like decluttering a room; it makes everything more organized and easier to work with. This step is crucial because it reduces the complexity of the equation, making it more straightforward to solve for x. When combining like terms, remember to pay close attention to the signs (positive or negative) in front of each term. These signs are part of the term and must be included in the calculation. Incorrectly combining like terms is a common mistake, so double-check your work here. Once we've combined the like terms, we're left with a much simpler equation that's ready for the next step: isolating the variable. This involves getting all the x-terms on one side of the equation and the constant terms on the other. We're on the home stretch now! Keep pushing, guys; you're doing great. The key here is to stay organized and methodical. Each step builds upon the previous one, so a clear understanding of each stage is vital.

Step 3: Move Variables to One Side

The next step in solving the equation is to move all the variable terms to one side of the equation. Our goal here is to get all the terms with 'x' on one side and the constant terms on the other. Looking at our simplified equation, -15x - 15 = -12x, we need to decide which side to move the x-terms to. It's often easier to move the term that will result in a positive coefficient for x, but either way will work. In this case, let’s move the -15x term from the left side to the right side. To do this, we add 15x to both sides of the equation. Remember, whatever we do to one side of the equation, we must do to the other to maintain balance. Adding 15x to both sides gives us: -15x - 15 + 15x = -12x + 15x. Simplifying this, the -15x and +15x on the left side cancel each other out, leaving us with just -15. On the right side, -12x + 15x simplifies to 3x. So, our equation now looks like this: -15 = 3x. We're getting closer to isolating x! This step is all about rearranging the equation to make it easier to solve. By moving the variable terms to one side, we're setting ourselves up to isolate x in the next step. The principle of maintaining balance in the equation is crucial here. It’s like a seesaw; if you add weight to one side, you must add the same weight to the other side to keep it level. This principle ensures that the equality remains true throughout our manipulations. Now that we have all the x-terms on one side, we’re just one step away from finding the value of x. We’ve successfully moved the variables, and the equation is looking much simpler. Keep up the momentum, guys! We're almost there. Remember to double-check your steps to ensure accuracy. A small mistake in an earlier step can throw off the entire solution.

Step 4: Isolate the Variable

We’re now at the final step: isolating the variable. Our equation currently looks like this: -15 = 3x. To isolate x, we need to get it all by itself on one side of the equation. Since x is being multiplied by 3, we need to perform the inverse operation, which is division. We will divide both sides of the equation by 3. This gives us: -15 / 3 = (3x) / 3. On the left side, -15 divided by 3 is -5. On the right side, the 3s cancel each other out, leaving us with just x. So, the equation simplifies to -5 = x or, more commonly written, x = -5. And there you have it! We've successfully solved the equation. This final step is where all our hard work pays off. Isolating the variable gives us the solution, the value of x that makes the equation true. Division is the key operation here, undoing the multiplication that was previously applied to x. Remember, we divide both sides by the same number to maintain the balance of the equation, just like we did when moving the variable terms. Once x is isolated, we have our answer. But it's always a good idea to check our solution to make sure it's correct. We can do this by plugging the value of x back into the original equation and seeing if both sides are equal. We'll do this in the next section. But for now, let's celebrate our success! We've navigated the equation step-by-step, and we've found the solution. Keep practicing, guys, and you'll become even more confident in your equation-solving skills. The ability to isolate variables is a fundamental skill in algebra and is used in countless mathematical contexts.

Step 5: Check Your Solution

It’s always a good idea to check your solution to ensure accuracy. We found that x = -5, so let’s plug this value back into the original equation: -5(1-5x) + 5(-8x-2) = -4x - 8x. Substituting x with -5, we get: -5(1 - 5(-5)) + 5(-8(-5) - 2) = -4(-5) - 8(-5). Now, let's simplify each side. On the left side: -5(1 + 25) + 5(40 - 2) = -5(26) + 5(38) = -130 + 190 = 60. On the right side: -4(-5) - 8(-5) = 20 + 40 = 60. Both sides equal 60, so our solution x = -5 is correct! Checking your solution is like proofreading a document; it catches any potential errors and ensures that your answer is accurate. This step is especially important in mathematics, where a small mistake can lead to a wrong solution. By plugging the value of x back into the original equation, we're verifying that our solution satisfies the equation. If the two sides are equal, we can be confident that our solution is correct. If they're not equal, it means we've made a mistake somewhere along the way, and we need to go back and review our steps. Checking your solution is also a great way to reinforce your understanding of the equation-solving process. It helps you see how the value of x works within the equation and confirms that our steps were logically sound. So, always make it a habit to check your solutions, guys. It’s a small investment of time that can save you from errors and build your confidence in your mathematical abilities. With our solution checked and verified, we can confidently say that we've successfully solved the equation! We've walked through each step, explained the logic behind it, and confirmed our answer. Great job, guys!

Conclusion

In this comprehensive guide, we've successfully navigated the equation -5(1-5x) + 5(-8x-2) = -4x - 8x, step by step. We started by distributing the numbers, then combining like terms, moving variables to one side, isolating the variable, and finally, checking our solution. Our final answer was x = -5. Mastering equation-solving is a fundamental skill in algebra, and by understanding each step, you’ll be well-equipped to tackle similar problems. Remember, practice is key! The more you work with equations, the more comfortable and confident you’ll become. Solving equations might seem challenging at first, but with a systematic approach and a clear understanding of the underlying principles, it becomes much more manageable. Each step we took had a purpose, from simplifying the equation to isolating the variable. And the final check ensures that our solution is accurate. We hope this guide has been helpful and has demystified the equation-solving process for you. Keep up the great work, guys, and don't be afraid to challenge yourself with more complex equations. The skills you've learned here will serve you well in your mathematical journey. And remember, math is not just about finding the right answer; it's about the process of problem-solving and critical thinking. So, embrace the challenge, stay curious, and keep learning. We're confident that with continued effort and practice, you'll master the art of solving equations and much more. Congratulations on making it to the end of this guide, and happy solving! We’ve covered a lot of ground, and you’ve demonstrated a great commitment to learning. Now, go forth and conquer those equations!