Solving 4x + 5y = 481: Find X And Y

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Solving for x and y in the Equation 4x + 5y = 481

Alright, let's dive into how to solve this equation: 4x + 5y = 481. This falls under the category of Diophantine equations, specifically a linear Diophantine equation, since we're looking for integer solutions for x and y. Because we only have one equation with two unknowns, there will be infinitely many solutions, but we can find a general form for them. Let's break it down step by step.

Initial Simplification

First, let's make sure our equation is clear. We have:

4x + 5y = 481

This is a linear equation where we want to find integer values for x and y that satisfy the equation. Linear Diophantine equations pop up frequently in number theory and algebra, and understanding how to tackle them is super useful. Often, real-world problems can be modeled using such equations, where you need integer solutions because you can't have fractions of items.

Finding a Particular Solution

To get started, we need to find at least one particular solution. One way to do this is by inspection or trial and error. Another approach is to use the Euclidean Algorithm to express the greatest common divisor (GCD) of 4 and 5 as a linear combination. Since 4 and 5 are relatively prime (their GCD is 1), we know solutions exist.

Let's try to find a solution by inspection. We want to find values for x and y such that 4x + 5y equals 481. Notice that 481 ends in 1, and 5y will always end in either 0 or 5. So, 4x must end in either 1 or 6 to make the sum end in 1. If 4x ends in 1, x would have to end in something like 0.25 or 0.75, which are not integers. So, 4x must end in 6. This means x must end in either 4 or 9.

Let’s try x = 4. Then, 4(4) + 5y = 16 + 5y = 481. So, 5y = 481 - 16 = 465, which gives y = 465 / 5 = 93. Voila! We found a solution: (x, y) = (4, 93).

General Solution

Now that we have one solution, we can find the general solution. If (xâ‚€, yâ‚€) is a particular solution to the equation ax + by = c, then the general solution is given by:

x = xâ‚€ + (b/gcd(a, b)) * n y = yâ‚€ - (a/gcd(a, b)) * n

where n is an integer.

In our case, a = 4, b = 5, c = 481, and our particular solution is (xâ‚€, yâ‚€) = (4, 93). The GCD(4, 5) = 1. So, our general solution is:

x = 4 + 5n y = 93 - 4n

where n is any integer. This means we can plug in any integer value for n and get a valid solution for x and y.

Verifying the General Solution

Let’s verify this by plugging the general solution back into the original equation:

4(4 + 5n) + 5(93 - 4n) = 16 + 20n + 465 - 20n = 481

The 20n terms cancel out, and we are left with 16 + 465 = 481, which is true. This confirms that our general solution is correct.

Examples of Solutions

Let's find a few more solutions by plugging in different values for n:

  • If n = 0: x = 4 + 5(0) = 4, y = 93 - 4(0) = 93 -> (4, 93)
  • If n = 1: x = 4 + 5(1) = 9, y = 93 - 4(1) = 89 -> (9, 89)
  • If n = -1: x = 4 + 5(-1) = -1, y = 93 - 4(-1) = 97 -> (-1, 97)
  • If n = 2: x = 4 + 5(2) = 14, y = 93 - 4(2) = 85 -> (14, 85)

Each of these pairs (x, y) will satisfy the equation 4x + 5y = 481.

Solving for Specific Constraints

Sometimes, you might have additional constraints, like requiring both x and y to be positive integers. In that case, we need to find the range of n that satisfies these conditions:

x = 4 + 5n > 0 y = 93 - 4n > 0

From the first inequality:

5n > -4 n > -4/5 n > -0.8

From the second inequality:

4n < 93 n < 93/4 n < 23.25

So, for both x and y to be positive integers, n must be an integer such that -0.8 < n < 23.25. This means n can be any integer from 0 to 23, inclusive.

In summary, guys, we found a particular solution to the Diophantine equation and then derived the general solution. We also looked at how to find solutions that meet certain constraints, like being positive integers. This approach can be used for any linear Diophantine equation with two variables.

Let's explore some more aspects of Diophantine equations, which can help you get a better handle on these types of problems. We'll look at the existence of solutions, methods to find them, and some common pitfalls.

Existence of Solutions

The Diophantine equation ax + by = c has integer solutions if and only if the greatest common divisor (GCD) of a and b divides c. That is, gcd(a, b) | c. If this condition is not met, no integer solutions exist. In our case, gcd(4, 5) = 1, and since 1 divides any integer, the equation 4x + 5y = 481 indeed has integer solutions.

Why is this the case? Well, if x and y are integers, then ax + by must be a multiple of gcd(a, b). If c is not a multiple of gcd(a, b), then ax + by cannot equal c. This principle is fundamental in determining whether a Diophantine equation has solutions.

Using the Euclidean Algorithm

The Euclidean Algorithm is a powerful tool for finding the GCD of two numbers and expressing it as a linear combination of those numbers. This is particularly useful for solving Diophantine equations. Let's apply it to our equation 4x + 5y = 481.

  1. Apply the Euclidean Algorithm to find gcd(4, 5):

    5 = 1 * 4 + 1 4 = 4 * 1 + 0

    So, gcd(4, 5) = 1.

  2. Express the GCD as a linear combination:

    From the first equation, 1 = 5 - 1 * 4. So, we can write 1 as 5(1) + 4(-1).

  3. Scale the equation to match the constant term:

    We want to solve 4x + 5y = 481. Since 1 = 5(1) + 4(-1), we multiply the entire equation by 481 to get:

    481 = 5(481) + 4(-481)

    Thus, one particular solution is x = -481 and y = 481.

This gives us a particular solution (xâ‚€, yâ‚€) = (-481, 481). Note that this is just one particular solution, and it's different from the one we found earlier (4, 93). The general solution, as we derived before, remains the same:

x = -481 + 5n y = 481 - 4n

Let's verify that this general solution is equivalent to the one we found before (x = 4 + 5n, y = 93 - 4n). If we set n' = n - 97, then:

x = -481 + 5(n' + 97) = -481 + 5n' + 485 = 4 + 5n' y = 481 - 4(n' + 97) = 481 - 4n' - 388 = 93 - 4n'

So, both general solutions are equivalent; they just start from different particular solutions.

Common Pitfalls

  1. Forgetting to Check for the Existence of Solutions: Always check if gcd(a, b) divides c before attempting to find solutions. If it doesn't, you're wasting your time!
  2. Incorrectly Applying the Euclidean Algorithm: Double-check each step to ensure you're finding the GCD correctly and expressing it as a linear combination accurately.
  3. Not Finding the General Solution: Finding one particular solution is just the beginning. You need to derive the general solution to represent all possible integer solutions.
  4. Ignoring Constraints: If there are additional constraints (e.g., x and y must be positive), remember to find the range of the parameter n that satisfies these constraints.
  5. Arithmetic Errors: Simple arithmetic mistakes can throw off your entire solution. Always double-check your calculations.

Practical Applications

Diophantine equations might seem abstract, but they have real-world applications in areas like:

  • Cryptography: They are used in certain cryptographic algorithms.
  • Computer Science: They can appear in problems related to integer programming.
  • Engineering: They can be used in problems involving discrete quantities.
  • Puzzle Solving: Many mathematical puzzles involve finding integer solutions to equations.

Understanding Diophantine equations can broaden your problem-solving toolkit and provide insights into various fields. Whether you're a student, engineer, or just a math enthusiast, mastering these concepts can be highly rewarding.

To wrap up, solving 4x + 5y = 481 involves finding a particular solution and then generalizing it to find all possible integer solutions. Remember to check for the existence of solutions, use the Euclidean Algorithm effectively, and watch out for common pitfalls. With practice, you'll become proficient in tackling these types of equations!