Solving For 2a + 5b + 3c Given A + B = 21 And B + C = 18

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Hey guys! Let's dive into a fun math problem today. We're going to figure out how to calculate the value of the expression 2a + 5b + 3c, but here's the twist: we don't have the individual values of a, b, and c. Instead, we're given two equations: a + b = 21 and b + c = 18. Sounds like a puzzle, right? Don't worry, we'll break it down step by step so it's super easy to follow. Math can be like a game if you know the rules, and we're about to learn the rules for this one!

Understanding the Problem

Before we jump into calculations, let’s make sure we understand what we're dealing with. We're given two equations:

  1. a + b = 21
  2. b + c = 18

Our goal is to find the value of the expression 2a + 5b + 3c. At first glance, it might seem impossible because we have three unknowns (a, b, and c) but only two equations. However, the key here is to manipulate the equations we have to get to the expression we want. We don't necessarily need to find the individual values of a, b, and c; we just need to find a way to combine them in the right way to get 2a + 5b + 3c. This is a classic algebraic trick, and it's super useful in many math problems. Think of it like this: we have building blocks (a + b and b + c), and we want to build a specific structure (2a + 5b + 3c). How do we arrange the blocks to get the structure we want? That's the challenge we're about to tackle. Understanding the problem is half the battle, and now we're ready to start strategizing our approach.

Strategy: Combining Equations

The core strategy here involves manipulating the given equations (a + b = 21 and b + c = 18) and combining them in a way that helps us get closer to the target expression (2a + 5b + 3c). The trick is to notice that we need multiples of a, b, and c. We have one a in the first equation, but we need two. We have two bs (one in each equation), but we need five. And we have one c in the second equation, but we need three. This suggests we might need to multiply the equations by different numbers and then add them together. It’s like we’re cooking up a recipe, and the equations are our ingredients. We need to figure out the right amounts of each ingredient to get the dish we want. So, let's start experimenting with multiplying and adding the equations. We’ll look for coefficients that, when combined, give us the 2a, 5b, and 3c we're aiming for. This might seem a bit like trial and error at first, but with a bit of algebraic intuition, we can find the right combination.

Step-by-Step Solution

Okay, let's get into the nitty-gritty of solving this. We need to find a clever way to combine our equations to match the expression 2a + 5b + 3c. Here's how we can do it:

  1. Multiply the first equation by 2:
    • If we multiply a + b = 21 by 2, we get 2a + 2b = 42. Notice that we've now got the 2a we need in our target expression. This is a great start! We're slowly building up the pieces of the puzzle.
  2. Multiply the second equation by 3:
    • Next, let's multiply b + c = 18 by 3. This gives us 3b + 3c = 54. Now we have the 3c we need, and we've also added more bs into the mix. This is good because we need a total of 5b in our final expression.
  3. Add the modified equations:
    • Now, let's add the two new equations we've created:
      • (2a + 2b) + (3b + 3c) = 42 + 54
      • This simplifies to 2a + 5b + 3c = 96

Boom! We've done it. By strategically multiplying and adding the equations, we've directly calculated the value of 2a + 5b + 3c. This is a neat trick, right? We didn't need to find the individual values of a, b, and c; we just manipulated the equations to get to the answer. It's like a mathematical magic trick! So, the answer is 96.

Verification and Alternative Approaches

Just to be extra sure, let's quickly verify our solution and think about if there might be other ways to solve this. Verification is always a good idea in math – it's like double-checking your work before you submit it. We've found that 2a + 5b + 3c = 96. Our steps were logical, and we carefully combined the equations, so it’s likely we’re correct. However, let's consider an alternative approach to solidify our understanding.

Another way we could think about this is by trying to express 2a + 5b + 3c in terms of the given expressions (a + b and b + c). We already know the values of these expressions, so if we can rewrite our target expression in terms of them, we’ll be golden. For example, we could rewrite 5b as 2b + 3b and then try to group the terms. We already used a similar method in our step-by-step solution, but recognizing different ways to approach the problem can help strengthen our problem-solving skills. It's like having multiple tools in your toolbox – each one might be better suited for a particular situation.

Common Mistakes to Avoid

When tackling problems like this, it's easy to make a few common mistakes. Let's chat about what they are so we can avoid them in the future. Knowing the pitfalls is just as important as knowing the steps to success. One frequent mistake is trying to solve for a, b, and c individually. While you could do that, it's more work than necessary and might lead to errors. Remember, the question doesn't ask for the individual values; it asks for the value of a specific expression. So, focusing on manipulating the equations directly is more efficient. Another common mistake is incorrectly distributing when multiplying the equations. For example, if you multiply a + b = 21 by 2, you need to make sure you multiply both a and b by 2, not just one of them. It’s the distributive property in action! Also, be careful with the signs when adding or subtracting equations. A small sign error can throw off the entire calculation. So, always double-check your work and make sure you're keeping track of those positives and negatives. By being aware of these common pitfalls, we can navigate these types of problems with more confidence and accuracy.

Practice Problems

Alright, guys, let's put our newfound skills to the test! Practice makes perfect, right? Here are a couple of problems similar to the one we just solved. Grab a pen and paper, and let's see if you can apply the same techniques to crack these:

  1. If x + y = 15 and y + z = 12, calculate 3x + 7y + 4z.
  2. Given p + q = 25 and q + r = 16, find the value of 2p + 5q + 3r.

Try to solve these problems without peeking at the solution. Remember our strategy: manipulate the equations by multiplying them by suitable constants and then add them together. Think about what coefficients you need to match the target expression. It's like figuring out the right ingredients for a recipe. Don't be afraid to experiment and try different approaches. And most importantly, have fun with it! Math can be like a puzzle, and the satisfaction of solving it is totally worth the effort. After you've given these a shot, you can compare your solutions and discuss your approaches. Let's get those math muscles flexing!

Conclusion

So, there you have it! We've successfully solved a tricky little math problem by combining equations in a clever way. The key takeaway here is that sometimes, in math (and in life!), you don't need to solve for every single variable to find the answer you're looking for. Instead, you can manipulate the information you have to directly get to the result. We learned how to multiply equations, add them together, and avoid common mistakes. It’s like we’ve added a new tool to our math toolbox, and that’s pretty awesome. Remember, practice is key. The more you work with these kinds of problems, the more comfortable and confident you’ll become. So, keep practicing, keep exploring, and keep enjoying the world of math! Who knows what other cool tricks and techniques you'll discover along the way? Keep those brains sharp, and happy calculating!