Finding Possible Values Of A In 3a + B = 21 A Step-by-Step Guide
Hey guys! Let's dive into a fun math problem today. We're going to explore the possible values of a variable in a simple equation. This is a common type of question you might encounter in math tests, and understanding how to solve it can really boost your confidence. So, grab your thinking caps, and let’s get started!
Understanding the Problem
The question we're tackling is this: We have two natural numbers, a and b, both less than 10. If 3a + b = 21, what are the possible values of a? We have some options to choose from, and our mission is to figure out which one is the right fit. This problem combines basic algebra with a bit of logical thinking, which makes it a great exercise for our brains. The key here is to systematically test the possibilities and see which ones hold true. Before we jump into the solution, let’s break down what each part of the problem means.
Decoding the Components
First, natural numbers are the positive whole numbers (1, 2, 3, and so on). This means we're not dealing with fractions, decimals, or negative numbers. This simplifies our task quite a bit! Next, we know that both a and b are less than 10. So, they can be any number from 1 to 9. This gives us a limited set of possibilities to work with. The equation 3a + b = 21 is the heart of the problem. It tells us that three times the value of a, plus the value of b, must equal 21. Our goal is to find which values of a satisfy this equation, given the constraints we just discussed. Finally, we have the multiple-choice options, which give us a guide for what the possible values of a could be. This helps us narrow down our search and makes the problem more manageable. Now that we've dissected the problem, let's move on to figuring out how to solve it.
Strategies for Solving
There are a couple of ways we can approach this problem. One method is to use substitution. We can try plugging in the values of a from the multiple-choice options and see if we can find a corresponding value for b that fits the criteria. If we substitute a value for a and find that b is a natural number less than 10, then that value of a is a valid solution. Another strategy is to rearrange the equation. We can rewrite 3a + b = 21 as b = 21 - 3a. This makes it easier to see how the value of b depends on the value of a. By plugging in different values for a, we can quickly calculate the value of b and check if it meets our conditions. Both of these methods are effective, and the best one for you might depend on your personal preference and how quickly you can do the calculations. In the next section, we'll apply these strategies to the problem and find the solution. Remember, the key is to be systematic and careful with your calculations. Let’s get to it!
Solving the Equation 3a + b = 21
Alright, let's put our strategies into action and crack this problem! We'll start by using the substitution method, plugging in the values of a from the multiple-choice options. This is a pretty straightforward way to see which values work. We'll go through each option one by one and check if they fit the equation 3a + b = 21, keeping in mind that both a and b must be natural numbers less than 10.
Testing the Options
Let's start with option A, which suggests that the possible values of a are 3, 5, and 6. First, we'll try a = 3. Plugging this into our equation, we get 3 * 3 + b = 21, which simplifies to 9 + b = 21. Subtracting 9 from both sides, we find that b = 12. But wait a minute! 12 is not less than 10, so a = 3 doesn't work. Next, let's try a = 5. Substituting this into the equation, we get 3 * 5 + b = 21, which simplifies to 15 + b = 21. Subtracting 15 from both sides, we find that b = 6. Okay, 6 is a natural number less than 10, so a = 5 is a possible solution. Now, let's try a = 6. Plugging this into the equation, we get 3 * 6 + b = 21, which simplifies to 18 + b = 21. Subtracting 18 from both sides, we find that b = 3. Again, 3 is a natural number less than 10, so a = 6 is also a possible solution. So far, we have two possible values for a: 5 and 6. But option A also includes 3, which we already ruled out. So, option A might not be the correct answer. Let's move on to option B and see what we find.
Evaluating Other Possibilities
Option B suggests that the possible values of a are 3, 6, and 9. We already know that a = 3 doesn't work because it leads to b = 12, which is greater than 10. We also know that a = 6 works, as it gives us b = 3. So, let's try a = 9. Plugging this into our equation, we get 3 * 9 + b = 21, which simplifies to 27 + b = 21. Uh oh! If we subtract 27 from both sides, we get b = -6. This is a negative number, which is not a natural number, so a = 9 doesn't work. Since option B includes 3 and 9, which are not valid solutions, we can rule out option B. Now, let's consider option C, which suggests that the possible values of a are 4, 5, and 6. We already know that a = 5 and a = 6 are possible solutions. So, let's try a = 4. Plugging this into our equation, we get 3 * 4 + b = 21, which simplifies to 12 + b = 21. Subtracting 12 from both sides, we find that b = 9. Great! 9 is a natural number less than 10, so a = 4 is also a possible solution. It looks like option C is a strong contender! Finally, let's check option D, which suggests that the possible values of a are 5, 6, and 9. We know that a = 5 and a = 6 work, but we already ruled out a = 9 because it leads to a negative value for b. So, option D is not the correct answer. After carefully testing each option, it seems like option C is the winner. But to be absolutely sure, let's summarize our findings and confirm our answer.
Determining the Correct Answer
Okay, guys, let's recap what we've found so far. We've systematically tested each of the multiple-choice options, plugging in the values of a and solving for b. We kept in mind that both a and b must be natural numbers less than 10. This process of elimination has helped us narrow down the possibilities and identify the correct answer.
Summarizing Our Findings
We started with option A, which included the values 3, 5, and 6 for a. We found that a = 3 didn't work because it resulted in b = 12, which is greater than 10. While a = 5 and a = 6 did work, the inclusion of 3 made option A incorrect. Next, we looked at option B, which included the values 3, 6, and 9 for a. Again, a = 3 didn't work, and a = 9 resulted in b = -6, which is not a natural number. So, we ruled out option B as well. Then, we examined option C, which included the values 4, 5, and 6 for a. We found that a = 4 resulted in b = 9, which is a valid solution. We already knew that a = 5 and a = 6 worked, so option C seemed promising. Finally, we checked option D, which included the values 5, 6, and 9 for a. We knew that a = 9 didn't work, so we ruled out option D. Based on our analysis, option C, which suggests that the possible values of a are 4, 5, and 6, seems to be the correct answer. To be completely certain, let's write out the solutions explicitly and confirm that they all satisfy the given conditions. When a = 4, 3a + b = 3 * 4 + 9 = 12 + 9 = 21. When a = 5, 3a + b = 3 * 5 + 6 = 15 + 6 = 21. And when a = 6, 3a + b = 3 * 6 + 3 = 18 + 3 = 21. All three values of a satisfy the equation, and the corresponding values of b are all natural numbers less than 10. So, we can confidently say that option C is the correct answer!
The Final Verdict
Therefore, the possible values of a are indeed 4, 5, and 6. We arrived at this solution by systematically testing the given options, applying our understanding of natural numbers and algebraic equations. This problem highlights the importance of careful calculation and logical reasoning in math. By breaking down the problem into smaller parts and using a step-by-step approach, we were able to find the answer with confidence. You guys did great! Keep practicing these skills, and you'll become math superstars in no time. Remember, math is not just about getting the right answer; it's about understanding the process and developing your problem-solving abilities. So, keep exploring, keep questioning, and keep learning!
Conclusion: Mastering Mathematical Problem-Solving
So, there you have it! We successfully navigated through this problem and found the possible values of a that satisfy the equation 3a + b = 21, given the constraints. This exercise wasn't just about finding the right answer; it was about honing our problem-solving skills and understanding the underlying concepts. By breaking down the problem, exploring different strategies, and carefully evaluating the options, we were able to arrive at the correct solution with confidence. Remember, math problems are like puzzles – they might seem daunting at first, but with the right approach and a bit of persistence, you can always find the solution. The key is to stay curious, keep practicing, and never be afraid to ask for help when you need it.
The Bigger Picture
Problems like this are not just academic exercises; they help us develop critical thinking skills that are valuable in all aspects of life. Learning to analyze information, identify patterns, and apply logical reasoning can help us make better decisions, solve real-world problems, and achieve our goals. So, the time and effort you invest in mastering mathematical concepts is an investment in your future success. Keep challenging yourselves with new problems, and don't be discouraged by setbacks. Every mistake is an opportunity to learn and grow. And remember, math can be fun! Approach it with a positive attitude, and you might be surprised at how much you enjoy the process of discovery.
Final Thoughts
I hope this explanation has been helpful and has given you a better understanding of how to approach problems like this. Remember, the journey of learning math is a marathon, not a sprint. Keep building your skills, keep exploring new concepts, and keep believing in yourselves. You guys have the potential to achieve great things! So, go out there and conquer the world of math, one problem at a time. And until next time, keep learning, keep growing, and keep shining! You've got this!
