Solving Math Problems: Finding 'a' When A(b+c)=36 & B+c=9

Hey guys! Let's dive into a fun math problem. We're given an equation, a * b + a * c = 36, and another piece of information: b + c = 9. Our mission, should we choose to accept it (and we totally do!), is to figure out the value of the natural number a. This kind of problem is super common in math, and understanding how to solve it can unlock a whole bunch of other problems. Trust me, it's like learning a secret code! In this article, we will break down the problem step by step. We'll explore different approaches and ensure you grasp the concept thoroughly. So, grab your pencils, get comfortable, and let’s crack this problem together. We will not only solve it, but we'll also make sure you understand why the solution works, which is way more important than just getting the right answer. This will become a piece of cake after we break it down! Are you ready?

Understanding the Problem: Unpacking the Given Information

Alright, before we start, let's make sure we're all on the same page. The problem gives us two key pieces of information. First, we have the equation a * b + a * c = 36*. This tells us that when you multiply a by b and a by c, and then add the results together, you get 36. It might look a little intimidating at first, but we will use some simple tricks. Second, we're told that b + c = 9*. This is a simpler equation, stating that the sum of b and c is equal to 9. This piece of information will be very important when solving the problem. Now, the beauty of math is that we can often simplify things to make them more manageable. This is the essence of problem-solving in math. We're looking for a natural number, meaning a whole number like 1, 2, 3, and so on. We can't have fractions or negative numbers here. These are our constraints. Having a clear understanding of the problem and the given information is the first and most crucial step in solving any math problem. It's like having the blueprint before you start building a house. Without the blueprint, you might end up with a crooked roof and doors that don't close! We must organize and understand the information to make the rest of the problem very easy. We are also very lucky to have two equations with a relationship between them.

Simplifying the Equation: The Power of Factoring

Okay, here's where things get interesting! Let's go back to our first equation: a * b + a * c = 36. Notice something? We have a multiplied by b and a multiplied by c. This is where factoring comes in handy. Factoring is basically like un-multiplying, or taking out a common factor. In this case, the common factor is a. When we factor out a from the equation, we get a(b + c) = 36. See? It's like we've grouped the terms together, making the equation much cleaner and easier to work with. This is a super useful technique in algebra and a lot of other areas of math. Think of it like this: instead of having two separate piles of apples (a * b* apples and a * c* apples), we've combined them into one pile, where a is how many groups of apples we have, and (b + c) is how many apples are in each group. By factoring, we've essentially rearranged the terms to reveal a hidden relationship. It is a fundamental concept in mathematics, allowing us to simplify complex expressions and equations. We transform a complicated equation into something much more manageable. Without this step, the problem would be significantly harder to solve. Learning how to factor opens up a whole new world of problem-solving possibilities in mathematics. Are you starting to see how the parts are fitting together?

Substituting Values: Unveiling the Solution

Now, remember the second piece of information we got? b + c = 9. Guess what? We can use this information to make our equation even simpler. We've already simplified and now we will use the second equation to help us. We know that (b + c) is equal to 9. And we also know that a(b + c) = 36. So, we can substitute the value of (b + c) in our equation. This is where the magic happens! Substituting means we take the value of something and put it in its place in the equation. So, in the equation a(b + c) = 36, we replace (b + c) with 9. This gives us a * 9 = 36. See how it's becoming simpler? The whole point of math is to simplify a problem so it can be easily solved. We've gone from a slightly complex equation to a very simple one. a times 9 equals 36. Now, we just need to find the value of a that makes this true. Think about what number multiplied by 9 equals 36. What do you think? Remember, we are looking for the natural number. We can use this value of (b + c) to simplify the equation and make it solvable. We are making progress! Are you with me?

Solving for a: Finding the Answer

Okay, we're almost there! We've simplified the equation to a * 9 = 36. Now it’s time to isolate a and find its value. To find the value of a, we need to get a by itself on one side of the equation. Remember, we're trying to find a number that, when multiplied by 9, gives us 36. To do this, we'll use the inverse operation of multiplication, which is division. We divide both sides of the equation by 9. This gives us (a * 9) / 9 = 36 / 9. On the left side, the 9s cancel out, leaving us with just a. On the right side, 36 divided by 9 equals 4. Therefore, a = 4. And there you have it! We've found our answer. The natural number a that satisfies both given equations is 4. Congrats! You solved it. You can now give yourself a pat on the back. Remember that you can always check your work. Now, let's reflect on the steps we took to reach the solution. This entire process gives you the foundation to tackle more complex problems. We started with two equations, we factored one of them to simplify it, and we used substitution to eliminate one of the variables. It will make other problems easier too.

Verification: Checking Your Work

It's always a good idea to verify your answer to ensure you've got it right. After you've found your solution, it is time to double-check. So, let's plug the value of a (which is 4) back into our original equations to see if they hold true. Remember the first equation? a * b + a * c = 36. Since a = 4, this becomes 4b + 4c = 36. Now, let's look at the second equation: b + c = 9. If we multiply both sides of this equation by 4, we get 4b + 4c = 36. This matches our first equation! This means our solution works. And, the more important thing is that you understand the why of the solution. Always remember that you can check your work to verify your answer. This is an important step that will help you build your confidence. It is always better to be sure, so don’t skip it. Now, you can move on to the next problem with total confidence.

Further Exploration: Expanding Your Skills

Well, that's it, guys! We've successfully solved the problem, and we've seen how to find the natural number a when given two equations. But, the fun doesn't stop here! What if we changed the numbers? What if instead of 36, we had 48 or 54? Would the approach change? Not really! The core principles of factoring and substitution would remain the same. This is the beauty of math, and once you learn the fundamentals, you will be able to tackle more difficult problems. This kind of problem is a great example of how different mathematical concepts connect and how you can use them to solve a problem. You can even try creating your own problems. Try to come up with your own equations and challenge your friends or classmates. The more you practice, the more confident you’ll become. Maybe you could try solving a similar problem but with a different number. You can also try to explore other mathematical concepts. Always push your limits. Learning math is like climbing a mountain; you need a good foundation to reach the summit! Keep practicing, keep exploring, and enjoy the journey. Congratulations on completing this problem! You have successfully learned how to solve for a, showing that you are well on your way to mastering mathematical problem-solving.