Solving (2x + 1)(5y - 3) = 30: Find X And Y

by ADMIN 44 views

Hey guys! Today, we're diving into a fun math problem where we need to find natural numbers x and y that satisfy the equation (2x + 1)(5y - 3) = 30. This might seem tricky at first, but with a bit of logical thinking and some number crunching, we can crack it! We'll break down the problem step by step, so you can follow along easily. This kind of problem is super helpful for building your problem-solving skills and getting more comfortable with number theory. So, let's get started and find those x and y values!

Understanding the Equation

Okay, so let's start by really understanding the equation we're dealing with: (2x + 1)(5y - 3) = 30. The core of this problem lies in figuring out what this equation is telling us and how we can use the given information to our advantage. First off, we know that x and y are natural numbers. What does that mean? Well, natural numbers are positive whole numbers – like 1, 2, 3, and so on. Zero is sometimes included, but for this problem, we'll focus on the positive integers.

Now, let's break down the equation itself. We have two expressions, (2x + 1) and (5y - 3), which are multiplied together, and the result is 30. This tells us that (2x + 1) and (5y - 3) must be factors of 30. A factor of a number is any whole number that divides evenly into that number. For example, the factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. We need to find pairs of these factors that could fit our expressions.

But there's a little twist! The expression (2x + 1) is always going to be an odd number. Why? Because 2 multiplied by any number is even, and adding 1 to an even number makes it odd. So, when we're looking at the factors of 30, we can immediately narrow down our options to only the odd factors. This is a key piece of information that simplifies the problem considerably. By understanding this constraint, we're not just blindly trying out every possible combination; we're using logic to focus on the most promising paths. This is a crucial skill in mathematics and problem-solving in general. By recognizing patterns and constraints, you can make complex problems much more manageable. This is a great first step in tackling this equation!

Identifying the Factors of 30

Alright, let's get down to the nitty-gritty and identify the factors of 30. As we discussed earlier, factors are numbers that divide evenly into 30. So, we need to figure out which numbers can be multiplied together to give us 30. We've already touched on this, but let's make a comprehensive list to be super clear. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30.

Now, remember the crucial point we made earlier? The expression (2x + 1) must be an odd number because 2 times any integer is even, and adding 1 makes it odd. This significantly narrows down our options. Looking at our list of factors, the odd factors of 30 are 1, 3, 5, and 15. These are the only numbers that (2x + 1) could possibly be equal to. This is a huge step forward because it means we can ignore the even factors (2, 6, 10, and 30) for this part of the equation.

So, why is this so important? Well, it's all about efficiency. Imagine trying to solve this problem without realizing that (2x + 1) has to be odd. You'd be testing a bunch of possibilities that are doomed to fail from the start! By recognizing this constraint, we're being smart problem-solvers. We're focusing our energy on the possibilities that actually have a chance of working. This is a fundamental principle in mathematics: look for patterns, identify constraints, and use them to simplify the problem.

Identifying these odd factors is like finding the right keys to unlock a door. We know that one of these numbers (1, 3, 5, or 15) must be the value of (2x + 1). Now, we can start testing these possibilities and see what happens with the other part of our equation, (5y - 3). This is where the fun really begins, as we start piecing together the solution. By pinpointing these specific factors, we've turned a potentially overwhelming task into a manageable one. This is the power of logical deduction in action!

Analyzing Possible Pairs

Okay, we've identified the odd factors of 30, which are 1, 3, 5, and 15. Now comes the fun part: analyzing possible pairs. We know that (2x + 1) has to be one of these odd factors. So, let's see what happens when we pair each of these with the corresponding factor that would multiply to 30. Remember, (2x + 1)(5y - 3) = 30. So, if we know what (2x + 1) is, we can figure out what (5y - 3) must be.

Let's break it down systematically:

  • If (2x + 1) = 1: Then (5y - 3) would have to be 30 (since 1 * 30 = 30). This is our first potential pair.
  • If (2x + 1) = 3: Then (5y - 3) would have to be 10 (since 3 * 10 = 30). This gives us our second pair.
  • If (2x + 1) = 5: Then (5y - 3) would have to be 6 (since 5 * 6 = 30). A third pair emerges!
  • If (2x + 1) = 15: Then (5y - 3) would have to be 2 (since 15 * 2 = 30). This is our final pair to investigate.

So, we've got four potential pairs of factors: (1, 30), (3, 10), (5, 6), and (15, 2). Each of these pairs represents a possible solution for our equation. But remember, we're not just looking for any numbers that multiply to 30; we're looking for natural numbers x and y that fit the expressions (2x + 1) and (5y - 3). That means we need to take each pair and see if we can actually find whole number values for x and y. This is where we'll put each pair to the test and see which ones hold up. It's like we've narrowed down our suspects, and now we need to investigate each one individually to find the culprit (or in this case, the correct values of x and y)!

Solving for x and y

Now that we have our potential pairs, it's time to solve for x and y in each case. Remember, we've got the following pairs to investigate:

  • (2x + 1) = 1 and (5y - 3) = 30
  • (2x + 1) = 3 and (5y - 3) = 10
  • (2x + 1) = 5 and (5y - 3) = 6
  • (2x + 1) = 15 and (5y - 3) = 2

Let's go through them one by one:

  1. Case 1: (2x + 1) = 1 and (5y - 3) = 30

    • Solving 2x + 1 = 1, we get 2x = 0, which means x = 0. But remember, x has to be a natural number, and 0 is not considered a natural number in this context. So, this case doesn't work.
    • We could also solve for y, but since x already failed, we can move on.

  2. Case 2: (2x + 1) = 3 and (5y - 3) = 10

    • Solving 2x + 1 = 3, we get 2x = 2, which means x = 1. This is a natural number, so it's a good sign!
    • Now let's solve 5y - 3 = 10. Adding 3 to both sides gives us 5y = 13. Dividing by 5, we get y = 13/5. This is not a natural number, so this case also doesn't work.

  3. Case 3: (2x + 1) = 5 and (5y - 3) = 6

    • Solving 2x + 1 = 5, we get 2x = 4, which means x = 2. Another natural number – great!
    • Now, let's solve 5y - 3 = 6. Adding 3 to both sides gives us 5y = 9. Dividing by 5, we get y = 9/5. Again, this is not a natural number, so this case is out too.

  4. Case 4: (2x + 1) = 15 and (5y - 3) = 2

    • Solving 2x + 1 = 15, we get 2x = 14, which means x = 7. Excellent, a natural number!
    • Now, let's solve 5y - 3 = 2. Adding 3 to both sides gives us 5y = 5. Dividing by 5, we get y = 1. Hooray! y is also a natural number.

We've finally found a solution that works! In the fourth case, x = 7 and y = 1. This means that the pair (7, 1) satisfies our original equation. All the hard work has paid off, and we've successfully navigated through the different possibilities to pinpoint the correct values for x and y. It's like we've unlocked the final puzzle piece and completed the picture!

Solution

Alright, after carefully analyzing all the possibilities, we've reached our solution! We found that the only pair of natural numbers (x, y) that satisfies the equation (2x + 1)(5y - 3) = 30 is x = 7 and y = 1. How cool is that? We started with a seemingly complex equation and, through a bit of logical thinking and systematic investigation, we were able to pinpoint the exact values that make it true.

So, the final answer is: x = 7 and y = 1*. This is a great example of how math isn't just about memorizing formulas and plugging in numbers. It's about understanding the underlying principles, identifying patterns, and using logic to solve problems. We used the concept of factors, recognized the constraint that (2x + 1) must be odd, and methodically tested each possibility until we found the right one. These are all valuable skills that extend far beyond the realm of mathematics.

What's particularly satisfying about this solution is that we didn't just guess and check. We used a structured approach, breaking the problem down into smaller, manageable steps. This is a crucial strategy for tackling any challenging problem, whether it's in math, science, or even everyday life. By taking a systematic approach, you can avoid getting overwhelmed and increase your chances of finding a successful solution. So, next time you're faced with a tough problem, remember the steps we took here: understand the problem, identify the key information, break it down into smaller parts, and systematically explore the possibilities. You might be surprised at what you can achieve!