Solving Algebraic Equations A Step-by-Step Guide To -13n - 53 = -2(2n - 14)

Introduction

Hey guys! Today, we're diving into solving a fun little algebraic equation. It's like a puzzle, and our goal is to figure out what value of 'n' makes the equation true. Specifically, we're tackling the equation -13n - 53 = -2(2n - 14). Don't worry if it looks a bit intimidating at first. We'll break it down step by step, so it'll all make sense. So, grab your pencils, notebooks, and let's get started! Remember, math is all about practice, and the more you practice, the better you'll get at it. We'll use the order of operations (PEMDAS/BODMAS) and algebraic manipulation to isolate the variable 'n' on one side of the equation. This involves distributing, combining like terms, and performing inverse operations to maintain the balance of the equation. By the end of this discussion, you'll not only know the solution but also understand the process of how to arrive at it. So, let's jump in and explore the world of algebra together!

Step 1: Distribute the -2 on the Right Side

The first thing we need to do is simplify both sides of the equation. On the left side, -13n - 53 is already as simple as it gets. But on the right side, we have -2(2n - 14). This means we need to distribute the -2 across the terms inside the parentheses. Remember, distribution means multiplying the term outside the parentheses by each term inside. So, we multiply -2 by 2n, which gives us -4n, and we multiply -2 by -14, which gives us +28. Be super careful with those negative signs; they can be sneaky! So, after distributing, our equation looks like this: -13n - 53 = -4n + 28. Now, both sides of the equation are a bit simpler, and we're one step closer to solving for 'n'. Distributing properly is key here, as it ensures that we're maintaining the equality of the equation. Think of it like balancing a scale – whatever you do to one side, you have to do to the other. And that's exactly what we're doing here, just simplifying each side to make it easier to work with.

Step 2: Combine Like Terms and Isolate 'n'

Okay, now that we've distributed and simplified, the next step is to get all the 'n' terms on one side of the equation and all the constant terms on the other side. This is like sorting your socks – you want all the pairs together, right? To do this, we can add 13n to both sides of the equation. Why 13n? Because it cancels out the -13n on the left side, leaving us with just the constant term. Adding 13n to both sides gives us: -53 = 9n + 28. See how the 'n' term has moved to the right side? Great! Now, let's get rid of that +28 on the right side. To do that, we subtract 28 from both sides. This gives us: -81 = 9n. We're almost there! We've got 'n' mostly isolated, but it's still being multiplied by 9. So, what's the opposite of multiplying by 9? Dividing by 9, of course! If you feel stuck, take a deep breath and remember the goal: isolate the variable. By adding 13n, we moved the variable terms to one side, and by subtracting 28, we isolated the variable term further. Each step brings us closer to the solution, and understanding the logic behind these steps is just as crucial as getting the right answer.

Step 3: Solve for 'n'

Alright, we've made it to the final step! We have -81 = 9n. To get 'n' all by itself, we need to divide both sides of the equation by 9. This is the inverse operation of multiplication, and it's what will finally reveal the value of 'n'. So, -81 divided by 9 is -9. And 9n divided by 9 is just 'n'. That means our solution is n = -9. Woohoo! We did it! We solved the equation! Now, it's always a good idea to check your answer, just to be sure. Plug -9 back into the original equation and see if both sides are equal. If they are, you know you've got the right answer. This final step is crucial because it confirms our solution and helps us build confidence in our problem-solving abilities. Think of it as the final piece of the puzzle fitting perfectly into place. By dividing both sides by 9, we successfully isolated 'n', revealing its value and completing our algebraic journey. Congratulations on making it this far!

Step 4: Checking the Solution

Okay, so we've found our solution: n = -9. But math isn't just about getting an answer; it's about being sure our answer is correct. That's where checking our solution comes in. It's like proofreading an essay or double-checking your work – it catches any mistakes we might have made along the way. To check our solution, we're going to plug -9 back into the original equation: -13n - 53 = -2(2n - 14). Replace every 'n' with -9, and let's see what happens. So, we get: -13(-9) - 53 = -2(2(-9) - 14). Now, we simplify each side following the order of operations. On the left side, -13 times -9 is 117. So, we have 117 - 53. On the right side, 2 times -9 is -18. So, we have -2(-18 - 14). Keep simplifying! 117 - 53 is 64. Inside the parentheses on the right, -18 - 14 is -32. So, we have -2(-32). And finally, -2 times -32 is 64. So, both sides of the equation equal 64! That means our solution, n = -9, is correct! Checking our solution not only confirms our answer but also reinforces our understanding of the equation and the steps we took to solve it. It's like a mini victory lap after a race, celebrating our success and solidifying our knowledge.

Conclusion

Alright, guys, we did it! We successfully solved the equation -13n - 53 = -2(2n - 14) and found that n = -9. We walked through each step, from distributing to combining like terms to isolating the variable, and we even checked our solution to make sure we were right. Solving equations like this is a fundamental skill in algebra, and it's something you'll use again and again in math and in other areas of your life. Remember, the key is to break down the problem into smaller, manageable steps. Don't be afraid of the equation; embrace it like a puzzle waiting to be solved. And most importantly, practice, practice, practice! The more you work through these types of problems, the more comfortable and confident you'll become. So, keep up the great work, and never stop exploring the fascinating world of mathematics! Remember, each equation solved is a step forward in your mathematical journey. The process we used – distributing, combining like terms, and isolating the variable – is a powerful tool in algebra, applicable to a wide range of equations. And by checking our solution, we ensure accuracy and build confidence in our skills. Keep practicing and exploring, and you'll be amazed at what you can achieve!