Solving For 'b': A Step-by-Step Guide

Hey guys! Ever get stuck trying to solve for a variable in an equation? No worries, it happens to the best of us. Today, we're going to break down a common type of problem: solving for ‘b’ in a linear equation. We'll use the equation 5b - 6 = 8 + 7b as our example. This might look intimidating at first, but trust me, we'll tackle it together, step by step. By the end of this guide, you'll be a pro at solving similar problems. The key to solving for a variable like ‘b’ is to isolate it on one side of the equation. This means we want to get ‘b’ all by itself on either the left or right side. To do this, we'll use algebraic manipulations, which basically means adding, subtracting, multiplying, or dividing both sides of the equation by the same value. This keeps the equation balanced and allows us to simplify things until we find the value of ‘b’. So, grab your pencils, and let's dive in! We'll go through each step meticulously, explaining the reasoning behind every move. Solving equations is a fundamental skill in mathematics, and mastering it will open doors to more advanced concepts. Plus, it's super satisfying when you finally get that ‘aha!’ moment and find the solution. Let's get started and turn those math challenges into triumphs!

Step 1: Gathering Like Terms

The first key step in solving for ‘b’ is to gather all the terms containing ‘b’ on one side of the equation and all the constant terms (the numbers without ‘b’) on the other side. This makes the equation much easier to work with. Think of it like organizing your closet – you want all your shirts together, all your pants together, and so on. In our equation, 5b - 6 = 8 + 7b, we have ‘b’ terms on both sides (5b and 7b) and constant terms on both sides (-6 and 8). To get the ‘b’ terms together, we can subtract 5b from both sides of the equation. Why subtract 5b? Because it will eliminate the ‘5b’ term on the left side, moving it to the right side. Remember, whatever we do to one side of the equation, we must do to the other side to keep it balanced. It’s like a see-saw – if you add weight to one side, you need to add the same weight to the other side to keep it level. So, subtracting 5b from both sides gives us: 5b - 6 - 5b = 8 + 7b - 5b. Now, simplify both sides. On the left side, 5b - 5b cancels out, leaving us with just -6. On the right side, 7b - 5b simplifies to 2b. Our equation now looks like this: -6 = 8 + 2b. Much cleaner, right? Next, we need to move the constant terms to the same side. We have -6 on the left and 8 on the right. To move the 8 to the left side, we subtract 8 from both sides of the equation. Again, we're doing this to isolate the ‘b’ term. Subtracting 8 from both sides gives us: -6 - 8 = 8 + 2b - 8. Simplify again. On the left side, -6 - 8 equals -14. On the right side, 8 - 8 cancels out, leaving us with just 2b. Now our equation looks like this: -14 = 2b. We're getting closer! By gathering like terms, we've simplified the equation significantly. The ‘b’ term is now almost isolated, and we just have one more step to completely solve for ‘b’.

Step 2: Isolating ‘b’

Now that we've gathered all the ‘b’ terms on one side and the constants on the other, the next crucial step is to isolate ‘b’ completely. Remember, our goal is to get ‘b’ all by itself, so we know its value. Looking at our current equation, -14 = 2b, we see that ‘b’ is being multiplied by 2. To undo this multiplication and isolate ‘b’, we need to perform the inverse operation, which is division. We'll divide both sides of the equation by 2. Why divide by 2? Because dividing 2b by 2 will leave us with just ‘b’. And remember, we must do the same thing to both sides to keep the equation balanced. So, let's divide both sides by 2: -14 / 2 = 2b / 2. Now, simplify. On the left side, -14 divided by 2 is -7. On the right side, 2b divided by 2 is simply b. This gives us our solution: -7 = b. Or, we can write it as b = -7. We did it! We've successfully isolated ‘b’ and found its value. This means that b equals -7. To double-check our work, we can substitute -7 back into the original equation and see if it holds true. This is a great way to make sure we haven't made any mistakes along the way. Substituting b = -7 into the original equation 5b - 6 = 8 + 7b gives us: 5(-7) - 6 = 8 + 7(-7). Simplify both sides: -35 - 6 = 8 - 49. -41 = -41. The equation holds true! This confirms that our solution, b = -7, is correct. Isolating the variable is a fundamental skill in algebra, and it's used in many different types of problems. By understanding this process, you'll be well-equipped to tackle more complex equations in the future. So, remember to perform the inverse operation to isolate the variable, and always double-check your answer by substituting it back into the original equation.

Step 3: Verifying the Solution

Alright, we've solved for ‘b’ and found that b = -7. But how do we know for sure if we got it right? This is where the important step of verifying the solution comes in. It's like checking your work on a test – you want to make sure you didn't make any silly mistakes along the way. To verify our solution, we'll substitute b = -7 back into the original equation: 5b - 6 = 8 + 7b. This means replacing every ‘b’ in the equation with -7. Let's do it: 5(-7) - 6 = 8 + 7(-7). Now, we simplify both sides of the equation, following the order of operations (PEMDAS/BODMAS). First, we perform the multiplications: On the left side: 5 * -7 = -35. On the right side: 7 * -7 = -49. Our equation now looks like this: -35 - 6 = 8 - 49. Next, we perform the additions and subtractions: On the left side: -35 - 6 = -41. On the right side: 8 - 49 = -41. So, our equation simplifies to: -41 = -41. What does this mean? It means that both sides of the equation are equal when we substitute b = -7. This is the magic moment! It confirms that our solution is correct. If the two sides of the equation didn't equal each other, it would mean we made a mistake somewhere in our calculations, and we'd need to go back and check our work. Verifying the solution is a crucial step in problem-solving because it gives us confidence in our answer. It's like having a final seal of approval on our work. Plus, it helps us catch any errors we might have made, preventing us from carrying those mistakes forward. So, always remember to verify your solutions, especially in math problems. It's a small step that can make a big difference in your accuracy and understanding. Now that we've verified our solution, we can confidently say that b = -7 is the correct answer to the equation 5b - 6 = 8 + 7b.

Conclusion

So, there you have it! We've successfully solved for ‘b’ in the equation 5b - 6 = 8 + 7b, and we've done it step-by-step, making sure we understand each process along the way. Let's recap the key steps we took: First, we gathered like terms, moving all the ‘b’ terms to one side of the equation and all the constant terms to the other side. This simplified the equation and made it easier to work with. Then, we isolated ‘b’ by performing the inverse operation of multiplication, which was division. This gave us the value of ‘b’. Finally, and most importantly, we verified our solution by substituting it back into the original equation. This gave us the confidence that our answer was correct. The solution we found was b = -7. By breaking down the problem into these manageable steps, we turned a potentially daunting equation into a solvable one. And that’s the beauty of algebra! It's all about breaking down complex problems into simpler parts and tackling them one at a time. Solving for variables like ‘b’ is a fundamental skill in mathematics, and it's used in many different areas, from simple equations to more advanced concepts. By mastering this skill, you're building a strong foundation for your mathematical journey. Remember, practice makes perfect. The more you solve equations like this, the more comfortable and confident you'll become. So, don't be afraid to tackle new problems and challenge yourself. And always remember to check your work and verify your solutions. Keep practicing, keep learning, and keep having fun with math! You've got this!