Solving Equations: Finding Natural Number Pairs

Oct 21, 2025 by SLV Team 48 views

Finding Natural Number Pairs: Unveiling Solutions to 3x + 6xy = 66

Hey guys! Let's dive into a classic math problem. We're going to explore how to find all pairs of natural numbers (that's positive whole numbers, for those who aren't familiar) that satisfy the equation 3x + 6xy = 66. This type of problem is super common in math competitions and even shows up in some university entrance exams. It's a great exercise in algebraic manipulation and logical thinking. The cool part? It's not as scary as it looks. We'll break it down step-by-step, making sure everyone can follow along. Ready? Let's do this!

Understanding the Problem and Setting Up

So, what exactly are we trying to do? We're looking for whole number values for x and y that, when plugged into the equation, make the left side equal to 66. Seems simple enough, right? The key here is recognizing the structure of the equation and how we can simplify it. The presence of both x and y in a term (6xy) suggests that factoring might be a good approach. Factoring is like the ultimate mathematical superpower – it lets us rewrite complex expressions into simpler, more manageable forms. It can really help us expose hidden relationships between the variables. We want to rearrange this equation to give us information about the possible values of x and y. The ultimate goal is to get the equation in a form where we can easily identify the possible values of x and y that satisfy the equation. This could involve looking at factors or isolating variables.

Here is our starting point: 3x + 6xy = 66. This equation is a linear Diophantine equation, which essentially means we're looking for integer solutions. Our first move is always to simplify the equation as much as possible, since it is always easier to solve simpler equations. Notice that all the terms (3x, 6xy, and 66) are divisible by 3. This means we can divide the entire equation by 3. Doing this simplifies things a lot, and it's a fundamental step that reduces the numbers we are dealing with.

So, after dividing by 3, we get: x + 2xy = 22. See? Already looking better. This also gives us: x(1 + 2y) = 22. Now, what's crucial here is recognizing that x and (1 + 2y) must be factors of 22. This is the core idea that unlocks the solution to the problem. We've transformed the initial equation into a form that's easier to work with. We can now use this relationship to find all possible values of x and y that will satisfy the equation. These factors are going to be whole numbers (because x and y are natural numbers), so we only have to consider the factor pairs of 22.

Factoring and Finding Possible Values

Okay, now that we've simplified our equation to x + 2xy = 22, the next crucial step is factoring. We want to manipulate the equation to express one side as a product of two terms. Think of it like this: if we can rewrite the equation as something like A * B = 22, then we know A and B must be factors of 22. This gives us a much more manageable set of possibilities to check. The factoring here is pretty straightforward. We'll factor out x from the left side of the equation. This is because x is a common factor in both terms on the left side (x and 2xy). Factoring out x gives us: x(1 + 2y) = 22. Voila! This equation is now in the form of a product, with x and (1 + 2y) being the two factors. Remember that x and y must be natural numbers (positive whole numbers). This means both x and (1 + 2y) must also be integers. Consequently, we need to identify all possible integer factor pairs of 22.

The factor pairs of 22 are: (1, 22), (2, 11), (11, 2), and (22, 1). Remember, since we are working with natural numbers, we only consider positive factors. This means that both x and (1 + 2y) must be positive. We will now examine each of these pairs to determine the values of x and y that they give rise to. The key here is to consider that (1 + 2y) will always be an odd number (because 2y is even, and adding 1 makes it odd). So, we can eliminate a few pairs right off the bat, since one of the factors must be odd. This crucial understanding streamlines our approach. Let's see how this works in practice.

Case 1: x = 1 and 1 + 2y = 22. Solving for y, we get 2y = 21, so y = 10.5. But y must be a natural number, so this case doesn't work. The value of y we have here, is not a whole number.
Case 2: x = 2 and 1 + 2y = 11. Solving for y, we get 2y = 10, so y = 5. This is a valid solution! We have our first pair of natural numbers (2, 5). We have found a solution to the equation.
Case 3: x = 11 and 1 + 2y = 2. Solving for y, we get 2y = 1, so y = 0.5. But y must be a natural number, so this case doesn't work. Remember, y must be greater than zero.
Case 4: x = 22 and 1 + 2y = 1. Solving for y, we get 2y = 0, so y = 0. But y must be a natural number, so this case also doesn't work. We are getting a similar problem here.

Identifying the Solutions

So, after carefully examining all the factor pairs of 22, we found only one valid solution. This is awesome! We've successfully navigated the equation. That takes us to our next step. Now we must consolidate the result.

The only pair of natural numbers (x, y) that satisfies the equation 3x + 6xy = 66 is (2, 5). This means that if we substitute x = 2 and y = 5 into the original equation, it will hold true. Let's do a quick check to confirm: 3(2) + 6(2)(5) = 6 + 60 = 66. Yep, it works perfectly! Finding this one solution is a testament to our systematic approach, from simplifying the equation to factoring and checking the possible values. This approach is key to solving similar problems. It's a great example of how mathematical tools like factoring can greatly simplify complex equations, making them much easier to solve. The process is not about luck; it's about applying a set of logical steps. Remember, the beauty of mathematics is in the journey of problem-solving. Each step provides deeper insights, and the satisfaction of reaching the solution is a reward in itself.

Conclusion and Key Takeaways

Alright, guys, we made it! We've successfully solved the equation 3x + 6xy = 66 and found the pair of natural numbers that satisfies it. We began with a seemingly complex equation and broke it down systematically, using algebraic manipulation and logical deduction. The whole process gave us a great understanding and appreciation for how equations work. Let's summarize what we've learned.

Simplification: Always look for opportunities to simplify the equation, such as dividing by a common factor. This is our first and most important tool.
Factoring: This is your secret weapon. By factoring, we transformed the equation into a form that exposed the relationships between the variables.
Factor Pairs: Understanding factor pairs is crucial. It lets you identify the possible values of the variables.
Logical Deduction: Analyzing the constraints (like natural numbers being positive) helps eliminate invalid solutions and find the right answer. These constraints help us a lot.

Solving this problem, and others like it, requires the consistent application of these basic tools. As you practice more, you will become more comfortable with algebraic manipulations and logical thinking. If you face similar problems, remember this framework. Break down the problem, factor, and think logically. Keep practicing, and you'll find that these types of problems become easier and more enjoyable. And as you solve more, you'll feel that incredible sense of accomplishment. That feeling is one of the best parts about working through math problems. So, keep up the great work, and keep exploring the amazing world of mathematics! Until next time, keep crunching those numbers!