Solving For B: A Step-by-Step Guide
Hey guys! Today, we're going to dive into solving a simple algebraic equation. Our main goal here is to find the value of 'b' that makes the equation true. Don't worry, it's not as scary as it sounds! We'll break it down step-by-step, so you can follow along easily. We're tackling the equation 15b - 40 = 5b + 120. Think of it like a puzzle where we need to isolate 'b' on one side of the equation. To do this, we'll use some fundamental algebraic principles. The beauty of algebra lies in its systematic approach. Each step we take is designed to simplify the equation while maintaining its balance. Remember, whatever we do to one side of the equation, we must do to the other. This ensures that the equality remains valid. Our journey starts with identifying the terms containing 'b' and the constant terms. The terms with 'b' are 15b and 5b, while the constants are -40 and 120. The aim is to group these like terms together. This makes the equation cleaner and easier to manipulate. We'll begin by moving the 5b term from the right side to the left side of the equation. To do this, we'll subtract 5b from both sides. This ensures that the 'b' terms are concentrated on one side, simplifying our next steps. Now, let's talk about why we subtract 5b from both sides. The fundamental principle here is maintaining balance. Imagine an old-fashioned weighing scale; if you remove weight from one side, you need to remove the same amount from the other to keep it level. Similarly, in our equation, subtracting 5b from both sides ensures that the equality remains intact. The equation now looks like this: 15b - 5b - 40 = 120. This simplification is crucial because it brings us closer to isolating 'b'.
Step-by-Step Solution
1. Combine Like Terms
Okay, so first things first, let's combine those like terms. We've got 15b on one side and 5b on the other. Our mission? To get all the bs on the same side. A simple way to do this is to subtract 5b from both sides of the equation. Think of it like this: what you do to one side, you gotta do to the other! This keeps everything balanced and fair in the equation world. By subtracting 5b from both sides, we're essentially neutralizing the 5b on the right side and moving its effect to the left side. This is a key step in isolating b. So, after subtracting 5b from both sides, our equation transforms. The right side becomes simpler, and the left side gets a bit more crowded with b terms, but in a manageable way. This step is all about organizing our equation to make it easier to solve. The new equation looks like this: 15b - 40 - 5b = 5b + 120 - 5b. Now, let's simplify it. On the left side, we can combine 15b and -5b. What does that give us? That's right, 10b. And on the right side, 5b - 5b cancels out, leaving us with just 120. So, our equation is now streamlined to: 10b - 40 = 120. See? We're making progress! We've successfully combined the b terms, and the equation is looking much cleaner. This simplification is crucial because it brings us one step closer to isolating b. Remember, each step we take is like peeling away a layer to reveal the solution. Now that we've combined the b terms, the next step is to deal with the constant terms. We're aiming to isolate b completely, and that means getting rid of anything that's hanging around with it on the left side of the equation.
2. Isolate the 'b' Term
Now, let's isolate that 'b' term. We've got 10b - 40 = 120. See that -40? We need to get rid of it. How? By doing the opposite – adding 40 to both sides! Adding 40 to both sides might seem like a small step, but it's a crucial move in our algebraic dance. It's like we're gently nudging the equation towards its solution. The goal here is to isolate the term with 'b', and getting rid of the constant term (-40) is a significant part of that process. Think of it like clearing the path for 'b' to stand alone and reveal its true value. Remember, whatever we add to one side, we have to add to the other. It's all about keeping the equation in balance, like a finely tuned seesaw. If we add 40 to the left side, we must also add 40 to the right side to maintain the equilibrium. This principle is fundamental to solving equations in algebra. So, when we add 40 to both sides, the equation transforms. The -40 on the left side is neutralized, and the right side gets a little bigger. This change is a deliberate step towards simplifying the equation further. By neutralizing the constant term on the left side, we're one step closer to having 'b' all by itself. This isolation is key to solving for 'b'. So, what happens when we add 40 to both sides? The equation becomes 10b - 40 + 40 = 120 + 40. Simplifying this, we get 10b = 160. Wow, that looks much simpler, right? We've successfully isolated the 10b term. We're on the home stretch now! This simplified equation is a testament to the power of methodical steps in algebra. We've carefully manipulated the equation, and now we're just a single step away from finding the value of 'b'. The hard work is paying off, and the solution is within reach. Now that we have 10b = 160, the next step is to get 'b' completely alone. This means we need to get rid of the 10 that's multiplying it. How do we do that? That's right, we divide!
3. Solve for 'b'
Alright, we're in the final stretch! We've got 10b = 160. To solve for 'b', we need to get 'b' all by its lonesome. And how do we do that when it's being multiplied by 10? Division is the answer! We're going to divide both sides of the equation by 10. Dividing both sides by 10 is the final piece of the puzzle. It's like the last turn of a Rubik's Cube that brings all the colors into alignment. This step is crucial because it directly reveals the value of 'b'. By dividing, we're undoing the multiplication that's currently keeping 'b' tied up. Think of it as liberating 'b' from its numerical constraint. Remember, consistency is key in algebra. What we do to one side, we must do to the other. Dividing both sides by 10 ensures that the equation remains balanced and that our solution is accurate. This principle of balance is a cornerstone of algebraic problem-solving. So, when we divide both sides by 10, the equation transforms. On the left side, the 10 that's multiplying b cancels out, leaving us with just 'b'. On the right side, 160 is divided by 10, giving us a numerical value. This transformation is the moment of truth, where the value of 'b' is revealed. The equation now reads 10b / 10 = 160 / 10. When we simplify this, we get b = 16. There it is! We've solved for b! It's like reaching the summit of a mountain after a challenging climb. The value b = 16 is the solution that makes the original equation true. We've navigated through the algebraic landscape, and we've arrived at our destination. Now that we've found the solution, it's always a good idea to check our work. This is like proofreading a document to ensure there are no errors. Checking our solution gives us confidence that we've done the problem correctly and that our answer is accurate.
Verification
Let's verify our answer. We found that b = 16. To make sure we're right, we'll plug this value back into our original equation: 15b - 40 = 5b + 120. If both sides of the equation are equal after substituting b = 16, then we know we've nailed it! Plugging in our solution is like testing a key in a lock. If the key fits and the lock opens smoothly, we know we've got the right key. Similarly, if substituting b = 16 makes the equation true, we know we've found the correct value for 'b'. This step is a crucial part of the problem-solving process. It's not just about finding an answer; it's about confirming that the answer is correct. Verifying our solution gives us peace of mind and reinforces our understanding of the equation. So, let's substitute b = 16 into the original equation. We get 15 * 16 - 40 = 5 * 16 + 120. Now, let's simplify each side. On the left side, 15 * 16 equals 240, so we have 240 - 40. On the right side, 5 * 16 equals 80, so we have 80 + 120. This substitution transforms the equation into a numerical expression that we can easily evaluate. It's like converting a foreign language into our native tongue so we can understand it clearly. By simplifying each side, we're checking to see if the two sides are equal. This is the heart of the verification process. If the two sides are indeed equal, then our solution is correct. So, what do we get when we simplify each side? On the left side, 240 - 40 equals 200. On the right side, 80 + 120 also equals 200. Bingo! Both sides are equal. This confirms that our solution, b = 16, is indeed correct. It's like the final puzzle piece clicking into place. The satisfaction of verifying our answer is a reward for the methodical steps we've taken. We've not only solved the equation, but we've also proven that our solution is accurate. This verification step strengthens our understanding of the problem and our confidence in our problem-solving skills. With the verification complete, we can confidently say that we've successfully solved for 'b'.
Final Answer
Therefore, the solution to the equation 15b - 40 = 5b + 120 is b = 16. Great job, guys! You've successfully navigated the world of algebra and found the value of 'b'. Remember, the key to solving equations is to take it step by step, keep things balanced, and always verify your answer. Keep practicing, and you'll become an equation-solving pro in no time! This journey through the equation has not only given us a numerical answer but also reinforced some fundamental algebraic principles. We've seen the importance of combining like terms, isolating variables, and maintaining balance in an equation. These are skills that extend beyond this particular problem and will serve you well in more advanced mathematical challenges. So, remember the steps we've taken today: combine like terms, isolate the variable, solve for the variable, and always verify your solution. These are the building blocks of algebraic problem-solving. As you continue to explore the world of mathematics, you'll encounter more complex equations and problems. But with a solid foundation in these basic principles, you'll be well-equipped to tackle them. Keep practicing, keep exploring, and never be afraid to ask questions. Mathematics is a journey of discovery, and every problem you solve is a step forward on that journey. And remember, guys, math isn't just about numbers and equations; it's about problem-solving, logical thinking, and perseverance. These are skills that are valuable in all aspects of life. So, embrace the challenge, enjoy the process, and celebrate your successes. You've got this!