Solving For X+y: X(x-y) = 5y-6 In Natural Numbers

Hey guys! Today, we're diving into a fun math problem that involves finding the values of x and y, which are natural numbers, that satisfy the equation x(x-y) = 5y - 6. Our ultimate goal is to figure out the value of x + y. So, let's put on our math hats and get started!

Understanding the Problem

Before we jump into solving, let's make sure we understand the problem clearly. We're given an equation, x(x - y) = 5y - 6, and we know that x and y are natural numbers. Remember, natural numbers are positive whole numbers (1, 2, 3, and so on). We need to find the values of x and y that make this equation true and then calculate the sum of those values. This involves algebraic manipulation, logical reasoning, and a bit of trial and error. Don't worry, we'll break it down step by step! Think of this as a puzzle where each step brings us closer to the solution. We're not just looking for any numbers; we're specifically looking for natural numbers, which adds a constraint that will help us narrow down the possibilities. We'll use this constraint to our advantage as we solve the equation. By understanding the type of numbers we're dealing with, we can eliminate many potential solutions right off the bat, making our task more manageable. Plus, knowing that x and y are positive will be crucial when we start rearranging the equation and looking for ways to isolate the variables. So, keep this in mind as we move forward: natural numbers are our key!

Rearranging the Equation

The first step in tackling this problem is to rearrange the equation to make it easier to work with. Our original equation is x(x - y) = 5y - 6. Let's expand the left side and then move all the terms to one side to see if we can simplify things. Expanding the left side gives us x² - xy = 5y - 6. Now, let's move all the terms to the left side: x² - xy - 5y + 6 = 0. This form of the equation is a bit cleaner, but it's still not immediately clear how to solve for x and y. We have x² term, an xy term, a y term, and a constant. It's a bit of a mixed bag! This is where we need to get creative and look for ways to factor or rearrange the equation further. One common strategy in these types of problems is to try to isolate one variable or to group terms in a way that allows us to factor. Factoring can be a game-changer because it transforms a complex equation into a product of simpler expressions, which are often easier to solve. So, keep factoring in the back of your mind as we move forward. We might not see an obvious way to factor just yet, but sometimes a little algebraic magic can reveal hidden structures. The goal here is to manipulate the equation into a form where we can apply techniques we know and love, like factoring or completing the square. Each step we take in rearranging the equation is like turning the pieces of a puzzle, trying to find the right fit.

Isolating and Factoring

Now, let's try to isolate y in our equation: x² - xy - 5y + 6 = 0. We can group the terms involving y together: x² + 6 = xy + 5y. Now, we can factor out y from the right side: x² + 6 = y(x + 5). This is a significant step! We've managed to isolate y on one side of the equation. To find y, we can divide both sides by (x + 5): y = (x² + 6) / (x + 5). This equation gives us a direct relationship between x and y. We know that both x and y are natural numbers, so this gives us a crucial piece of information. The expression (x² + 6) / (x + 5) must result in a natural number. This means that (x² + 6) must be divisible by (x + 5). This divisibility condition is the key to unlocking the solution! We can use this to test different values of x and see which ones give us a natural number for y. Alternatively, we can use polynomial division to simplify the expression further. Polynomial division might sound intimidating, but it's just a way to divide polynomials, similar to how you divide regular numbers. By dividing (x² + 6) by (x + 5), we can rewrite the expression in a more manageable form. This will help us better understand the relationship between x and y and make it easier to find values that satisfy the condition that y is a natural number. Remember, our goal is to find natural number solutions, so any technique that helps us identify those solutions is worth exploring. Keep this divisibility rule in mind as we proceed; it's our guiding star in this mathematical journey!

Using Polynomial Division (or a Clever Trick!)

Let's use polynomial division (or a clever trick!) to simplify the expression y = (x² + 6) / (x + 5). We can rewrite x² + 6 as x² - 25 + 31. Why did we do this? Because x² - 25 is a difference of squares and can be factored! So, we have y = (x² - 25 + 31) / (x + 5). Now, we can rewrite x² - 25 as (x + 5)(x - 5). This gives us y = [(x + 5)(x - 5) + 31] / (x + 5). We can now split the fraction: y = (x + 5)(x - 5) / (x + 5) + 31 / (x + 5). Simplifying the first term, we get y = (x - 5) + 31 / (x + 5). This is a fantastic result! We've transformed the expression into something much easier to work with. We now have y expressed as the sum of an integer (x - 5) and a fraction (31 / (x + 5)). Since y is a natural number, both (x - 5) and 31 / (x + 5) must be integers. This means that (x + 5) must be a factor of 31. Remember, 31 is a prime number, which means its only factors are 1 and 31. This significantly narrows down the possibilities for (x + 5). This clever trick of rewriting the numerator and splitting the fraction is a powerful technique in problem-solving. It allows us to transform a complex expression into a simpler one, revealing hidden relationships and making it easier to find solutions. So, always be on the lookout for ways to manipulate expressions algebraically; you never know when you might stumble upon a breakthrough!

Finding Possible Values for x

Now that we know (x + 5) must be a factor of 31, let's find the possible values for x. Since 31 is prime, its only factors are 1 and 31. So, we have two possibilities: x + 5 = 1 or x + 5 = 31. If x + 5 = 1, then x = -4. But x must be a natural number, so this solution doesn't work. If x + 5 = 31, then x = 26. This is a valid solution since 26 is a natural number! So, we've found a possible value for x: x = 26. Remember, the fact that 31 is a prime number was crucial here. It severely limited the possible factors and made our job much easier. If 31 had more factors, we would have had more cases to consider. This highlights the importance of recognizing the properties of numbers in problem-solving. Prime numbers have special characteristics that can be very helpful in simplifying problems. In this case, the primality of 31 led us directly to a potential value for x. This is a great example of how number theory concepts can come into play in algebraic problems. By understanding the building blocks of numbers (like primes), we can often find elegant solutions to seemingly complex equations. So, always keep your number theory knowledge sharp; it might just be the key to unlocking the next math puzzle!

Calculating y

Now that we have a value for x (x = 26), let's calculate the corresponding value for y. We can use the equation y = (x² + 6) / (x + 5) or the simplified form y = (x - 5) + 31 / (x + 5). Let's use the simplified form; it's easier to compute. Substituting x = 26, we get y = (26 - 5) + 31 / (26 + 5) = 21 + 31 / 31 = 21 + 1 = 22. So, when x = 26, y = 22. We now have a pair of values (x = 26, y = 22) that satisfy the equation. But before we jump to conclusions, we need to make sure that these values actually work in the original equation. It's always a good idea to check our solutions, especially in problems like this where we've done a bit of algebraic manipulation. Plugging the values back into the original equation helps us catch any potential errors we might have made along the way. Think of it as a final quality check to ensure our answer is correct. This step is crucial in preventing careless mistakes and giving us confidence in our solution. So, let's take a moment to verify that x = 26 and y = 22 indeed satisfy the equation x(x - y) = 5y - 6.

Verifying the Solution

Let's verify our solution by plugging x = 26 and y = 22 into the original equation: x(x - y) = 5y - 6. Substituting the values, we get 26(26 - 22) = 5(22) - 6. Simplifying the left side, we have 26(4) = 104. Simplifying the right side, we have 110 - 6 = 104. Since both sides are equal (104 = 104), our solution is correct! We've confirmed that x = 26 and y = 22 satisfy the original equation. This gives us a great sense of accomplishment! We've successfully navigated through the algebraic manipulations, the divisibility conditions, and the potential pitfalls to arrive at a verified solution. This step of verification is not just a formality; it's a crucial part of the problem-solving process. It reinforces our understanding of the problem and gives us the confidence to move on to the final step: calculating x + y. So, let's take a moment to appreciate the journey we've taken so far and prepare for the grand finale!

Calculating x + y

Finally, let's calculate x + y. We found that x = 26 and y = 22. So, x + y = 26 + 22 = 48. Therefore, the value of x + y is 48. We did it! We started with a seemingly complex equation, used our algebraic skills and number theory knowledge, and arrived at the solution. This is a testament to the power of persistence and careful problem-solving. Remember, math problems are often like puzzles; they require us to think creatively, try different approaches, and never give up. And the satisfaction of solving a challenging problem is truly rewarding! So, let's take a moment to celebrate our success and appreciate the beauty of mathematics.

Conclusion

In conclusion, if natural numbers x and y satisfy the equation x(x - y) = 5y - 6, then x + y = 48. This problem was a great exercise in algebraic manipulation, factoring, and using divisibility rules. I hope you guys enjoyed this walkthrough! Remember, math is all about practice and perseverance. Keep exploring, keep solving, and keep having fun! You've shown that by breaking down a complex problem into smaller, manageable steps, we can tackle even the most challenging mathematical puzzles. So, keep that problem-solving spirit alive and keep exploring the fascinating world of mathematics! Until next time, happy calculating! This type of problem appears frequently in math competitions, so mastering the techniques we used here will definitely give you an edge. Keep practicing similar problems, and you'll become a math whiz in no time!