Solving For X: 2x + 3(x + A + B) = 66, A + B = 17

Hey guys! Today, we're diving into a fun little math problem that combines algebra with some basic arithmetic. We've got the equation 2x + 3 * (x + a + b) = 66, and we know that a + b = 17. Our mission, should we choose to accept it (and we totally do!), is to find the value of x. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into solving, let's break down what we're dealing with. This problem is a classic example of an algebraic equation where we need to isolate the variable 'x'. We have a couple of key pieces of information: the main equation, 2x + 3 * (x + a + b) = 66, and the additional fact that a + b = 17. The presence of 'a' and 'b' might seem a bit daunting at first, but don't worry, we've got this! The beauty of math is that we can use the information we have to simplify and solve. In this case, the value of a + b = 17 is our secret weapon. By substituting this value into the main equation, we can eliminate 'a' and 'b' and end up with an equation that only involves 'x'. This is a common strategy in algebra: look for ways to reduce the complexity of the equation by substituting known values or simplifying expressions. Remember, the goal is always to get 'x' by itself on one side of the equation so we can see what it equals. Once we've done that, we've cracked the code!

Step-by-Step Solution

Okay, let's get down to business and solve this equation step by step. Here's how we'll do it:

1. Substitute the value of a + b: We know that a + b = 17, so let's plug that into our main equation. This gives us: 2x + 3 * (x + 17) = 66.
2. Distribute the 3: Next, we need to get rid of those parentheses. We do this by distributing the 3 across the terms inside: 2x + 3x + 51 = 66. Remember, distributing means multiplying the term outside the parentheses by each term inside.
3. Combine like terms: Now, let's simplify things by combining the 'x' terms: 2x + 3x becomes 5x. So our equation now looks like this: 5x + 51 = 66. Combining like terms is a fundamental step in solving algebraic equations, making the equation easier to work with.
4. Isolate the x term: We want to get the 5x term by itself on one side of the equation. To do this, we subtract 51 from both sides: 5x + 51 - 51 = 66 - 51. This simplifies to 5x = 15. Remember, whatever we do to one side of the equation, we must do to the other to keep things balanced.
5. Solve for x: Finally, to find x, we divide both sides of the equation by 5: 5x / 5 = 15 / 5. This gives us the solution: x = 3.

And there you have it! We've successfully solved for x. It might seem like a lot of steps, but each one is straightforward and brings us closer to the answer. Practice makes perfect, so the more you solve these types of problems, the easier they'll become.

Checking Our Answer

Alright, we've found that x = 3, but how do we know if we're right? It's always a good idea to check our answer, guys! This not only confirms that we haven't made any mistakes but also helps solidify our understanding of the problem. To check our answer, we're going to plug x = 3 back into the original equation and see if it holds true. Remember the original equation? It was 2x + 3 * (x + a + b) = 66. We also know that a + b = 17. So, let's substitute x = 3 and a + b = 17 into the equation:

2(3) + 3 * (3 + 17) = 66

Now, we just need to simplify and see if both sides of the equation are equal. First, let's do the multiplication within the parentheses: 3 + 17 = 20. So the equation becomes:

2(3) + 3 * (20) = 66

Next, we perform the multiplications: 2 * 3 = 6 and 3 * 20 = 60. Now we have:

6 + 60 = 66

Finally, we add the numbers on the left side: 6 + 60 = 66. And guess what? The equation holds true! 66 = 66. This means our answer, x = 3, is correct. Checking our answer is a crucial step in problem-solving. It gives us confidence in our solution and helps us catch any errors we might have made along the way. So, always take a few extra minutes to check your work – it's totally worth it!

Why This Matters: Real-World Applications

You might be thinking, "Okay, cool, we solved for x, but when am I ever going to use this in real life?" That's a fair question! While you might not be solving algebraic equations every day, the skills you develop in doing so are incredibly valuable. These types of problems aren't just about numbers and symbols; they're about critical thinking, problem-solving, and logical reasoning. These are skills that you'll use in all sorts of situations, from managing your finances to making important decisions at work. For example, let's say you're planning a budget. You have a certain amount of money coming in, and you have various expenses. You can use algebraic thinking to figure out how much you can spend on each category while still staying within your budget. This might involve setting up an equation similar to the one we just solved, where the variables represent different expenses. Or, imagine you're working on a project at work. You have a deadline, a team of people, and a set of tasks to complete. You can use problem-solving skills to break the project down into smaller steps, allocate resources effectively, and track your progress. This might involve identifying constraints, finding the most efficient way to complete tasks, and adjusting your plan as needed. The ability to think logically and systematically is essential in almost any field, from science and engineering to business and the arts. So, while solving for x might seem abstract, the underlying skills are incredibly practical and will serve you well throughout your life. Keep practicing, and you'll be amazed at how these skills translate into success in various areas!

Common Pitfalls and How to Avoid Them

Alright, guys, let's talk about some common mistakes people make when solving problems like this and how we can avoid them. Solving equations can be tricky, and it's easy to slip up if you're not careful. But don't worry, we're going to go over some of the biggest pitfalls so you can steer clear of them!

One of the most common mistakes is messing up the order of operations. Remember PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction)? It's crucial to follow this order when simplifying expressions. For example, in our equation, we needed to distribute the 3 before we could combine like terms. If we had added 2x and 3 first, we would have gotten the wrong answer. To avoid this, always double-check that you're following PEMDAS correctly. Another common mistake is forgetting to distribute properly. When you have a number multiplying a set of terms inside parentheses, you need to multiply each term inside the parentheses by that number. For instance, in our equation, we had 3 * (x + 17). We needed to multiply both the x and the 17 by 3. Some people might forget to multiply the 17, which would lead to an incorrect solution. To avoid this, make sure you distribute carefully and double-check your work. Sign errors are another frequent culprit. It's easy to mix up positive and negative signs, especially when you're dealing with multiple terms. For example, when we subtracted 51 from both sides of the equation, we needed to make sure we subtracted it correctly from both sides. A simple sign error can throw off the entire solution. To avoid sign errors, write out each step clearly and pay close attention to the signs of the numbers. Finally, not checking your answer is a big mistake. We talked about this earlier, but it's worth repeating. Checking your answer is the best way to catch any errors you might have made. It's like having a safety net that ensures you've arrived at the correct solution. So, always take the time to plug your answer back into the original equation and see if it works. By being aware of these common pitfalls and taking steps to avoid them, you'll become a much more confident and accurate problem solver. Remember, practice makes perfect, so keep working at it!

Practice Problems

Okay, guys, now it's your turn to shine! To really solidify your understanding of solving equations like this, let's tackle a few practice problems. The best way to learn math is by doing, so grab a pencil and paper, and let's get to work!

Here are a couple of problems similar to the one we just solved:

1. If c + d = 22, calculate the natural number y from the expression 4y + 2 * (y + c + d) = 88.
2. Given p + q = 15, find the value of z in the equation 3z + 5 * (z + p + q) = 111.

These problems follow the same basic structure as the one we worked through together. You'll need to use substitution, distribution, combining like terms, and isolating the variable to find the solution. Remember to follow the steps we outlined earlier, and don't forget to check your answers!

To make the most of these practice problems, try to solve them on your own first. If you get stuck, don't be afraid to look back at the steps we took in the example problem. And if you're still struggling, that's okay too! Math can be challenging, and it's normal to need some help along the way. You can ask a friend, a teacher, or even search online for explanations and examples.

The key is to keep practicing and to not give up. The more problems you solve, the more comfortable you'll become with the process. And the more comfortable you are, the easier it will be to tackle even more complex problems in the future. So, go ahead, give these practice problems a try. You've got this!

Conclusion

So, there you have it, guys! We've successfully navigated the world of algebraic equations and learned how to solve for x when given an equation like 2x + 3 * (x + a + b) = 66 and the value of a + b. We started by understanding the problem, then we broke it down into manageable steps: substituting, distributing, combining like terms, isolating x, and finally, solving for x. We even learned the importance of checking our answer to make sure we didn't make any mistakes. But more than just solving a specific problem, we've explored some valuable problem-solving skills that you can use in all sorts of situations. Understanding the steps we took and the reasoning behind them is key to building your math confidence and tackling future challenges. Remember, math isn't just about memorizing formulas and procedures; it's about thinking logically and creatively. And these are skills that will serve you well in all aspects of your life. So, keep practicing, keep exploring, and keep challenging yourself. You've got the tools, the knowledge, and the determination to succeed. Now go out there and conquer those equations! You're math superheroes in the making!