Solving Equations: Find A+B+C With A+2B+C=51 & 2A+B+2C=42

Hey everyone! Let's dive into a cool math problem. We've got a system of equations, and our mission, should we choose to accept it, is to figure out the value of A + B + C. We're given two equations: A + 2B + C = 51 and 2A + B + 2C = 42. Don't worry, it might seem a little intimidating at first, but we'll break it down step by step. The goal is to find a way to manipulate these equations so that we can isolate and calculate A + B + C. It's like a puzzle, and we're the puzzle solvers! So, grab your pencils, and let's get started. This problem is a classic example of how algebra can be used to solve real-world problems, even if those "real-world problems" are just mathematical equations. The key here is to use the given information to work our way towards the solution. The steps involved will not only show us the answer, but they also provide a deeper understanding of how equations work. Let's make this math adventure fun and educational.

Understanding the Problem: The Basics

Okay, first things first, let's make sure we totally understand what we're dealing with. We have two equations, which are essentially two statements that relate three unknown variables: A, B, and C. Each equation represents a relationship between these variables. Our goal is to find the value of a specific combination of these variables: A + B + C. To do this, we'll need to combine the given equations strategically. We can't just guess the values of A, B, and C individually, as there are many possible solutions that could satisfy either equation on its own. That's why we need to use both equations together. The beauty of this is that the solution, when found, will give us a unique value for A + B + C. So, even though we don't know the individual values of A, B, and C, we can still determine the value of their sum. This is the core concept. When we solve for A + B + C, we are essentially finding a specific point that satisfies both equations simultaneously. This is very important as the foundation for how we will approach and solve the problem at hand. Without this understanding, we would be missing a core element of the math and the solution itself.

The Strategic Approach: Combining the Equations

Now, let's get to the exciting part: the strategy. Our aim is to manipulate the given equations, so we create an expression that directly gives us A + B + C. There are several ways to do this, but one of the most straightforward is to combine the two equations in such a way that the coefficients of A, B, and C add up to 1. Let's think about how we can achieve this. Looking at our equations, we have: Equation 1: A + 2B + C = 51 and Equation 2: 2A + B + 2C = 42. If we can somehow get the coefficients of A, B, and C to become 1 in each term when we add them together, we will have A + B + C on the left side of the equation, and the right side will be a single numerical value which will be the result. This is our target, and we can use a variety of methods to reach it. So, we can add multiples of the equations, subtract one from the other, or even divide the equations by a number. In this instance, the method is not that complicated. Let's think through this process and find the best way to work through the math. With a little manipulation, we will reach our goal, and we will have our solution!

The Solution Unveiled: The Step-by-Step Breakdown

Alright, let's get down to business and see how we can crack this equation. Here's the step-by-step breakdown:

1. Adding the Equations The best strategy here is to first add the two equations together. So, (A + 2B + C) + (2A + B + 2C) = 51 + 42. This simplifies to 3A + 3B + 3C = 93.
2. Simplifying the Equation Now, we have 3A + 3B + 3C = 93. Notice that all the coefficients (3) are the same. This means we can divide the entire equation by 3 to simplify it further. This gives us (3A + 3B + 3C) / 3 = 93 / 3. Simplifying, we get A + B + C = 31.
3. The Answer And there you have it! We have successfully found the value of A + B + C. Therefore, A + B + C = 31.

So, it’s a simple but elegant solution. We didn't have to solve for the individual values of A, B, or C. Instead, we cleverly combined the equations to directly find the sum we were looking for. This is a powerful technique in algebra, demonstrating that sometimes, you don't need to know everything to find the answer to a specific problem.

Why This Works: The Underlying Principle

Let's pause and think about why this strategy worked. We essentially took advantage of the distributive property of multiplication and division. When we added the equations, we kept the balance, which is a fundamental principle in algebra. Whatever we did to one side of the equation, we did to the other side. This is crucial because it ensures that the equality holds true throughout the entire process. Then, by dividing by 3, we were able to isolate the sum A + B + C. The reason this works is because we are working with linear equations. This means that the variables are raised to the power of 1, and there are no more complex operations like squares or cubes. This linearity makes it easier to manipulate the equations and find a solution by adding, subtracting, multiplying, or dividing. The underlying principle is that as long as we perform valid algebraic operations, the solution will remain constant. The key is to maintain the equation's balance at every step, making it easier to reach the desired result.

Additional Insights and Tips: Refining Your Skills

This type of problem is a great exercise in algebraic manipulation. Here are a few more tips and insights that can help you become a better problem solver:

• Practice Makes Perfect: The more you practice solving these types of problems, the more familiar you'll become with the different strategies and techniques. Try solving similar problems with different equations and variables. The more you practice, the better you will be.
• Look for Patterns: Recognize the patterns in the coefficients. In this problem, the coefficients of A, B, and C in the simplified equation were all the same, making it easy to divide. Learn to notice these patterns, so you can devise your own solution strategies.
• Explore Different Methods: There's often more than one way to solve an algebraic problem. Experiment with different approaches to see which one works best for you. For example, you could try to isolate one variable and substitute it into the other equation. There is never a single, correct way to solve these equations. Experiment and enjoy the process.
• Double-Check Your Work: Always double-check your calculations to avoid silly mistakes. It's easy to make a mistake when you're working with multiple steps. Once you arrive at the final answer, go back through the process and make sure you can explain what you did and why.

Conclusion: Mastering the Equations

So, there you have it! We've successfully solved the problem and found that A + B + C = 31. We hope that this detailed explanation has helped you better understand how to tackle this type of algebraic problem. Remember, algebra is all about the relationships between variables and using those relationships to solve problems. With practice and a little bit of strategic thinking, you can master these skills and conquer any equation that comes your way. Keep practicing, stay curious, and don't be afraid to experiment. The more you do, the more confident you'll become. Remember to always look for those patterns and keep that mental map of how to approach different problems. Keep up the great work, and we will see you in the next math adventure! Now, go forth and apply what you've learned. You are well on your way to becoming a math master!