How Do I Find All Triplets of Positive Integers for the Given System of Equations?

The challenge of finding all triplets of positive integers ((x, y, z)) that satisfy the following system of equations:

xy - x - z 1 (Equation A) xy y - z 2006 (Equation B)

provides a rich ground for analytical exploration, bypassing brute force methods. This guide will walk you through a detailed, systematic approach to solving these equations.

Understanding the Equations

Let us start by understanding the provided system of equations:

xy - x - z 1 (Equation A) xy y - z 2006 (Equation B)

These two equations involve three variables, making it evident that there are multiple solutions. Our primary goal is to find methods to eliminate one variable, allowing us to express the other two in a manner that helps us identify possible integer values for all variables.

A Direct Approach to Eliminate One Variable

By adding Equations A and B, we get a new equation:

2xy - x y 2007 (Equation C)

This equation eliminates (z) and simplifies the problem. Our next step is to manipulate Equation C to facilitate the identification of integer solutions.

Manipulating the Equation

Rearranging Equation C, we have:

x 2y 1 - y 2007

Which simplifies to:

x y 2007 - 1 (Equation D)

Multiplying both sides by 2, we get:

2x 4014 - 2y (Equation E)

Applying componendo on both sides of Equation E, we obtain:

2x 1 4015 / (2y 1) (Equation F)

This equation can now be used to find suitable values of (y), allowing us to determine (x) and subsequently (z).

Determining Suitable Values for (y)

We know that 4015 5 * 11 * 73. The factors of 4015 are: 1, 5, 11, 55, 73, 365, 803, and 4015. We equate (2y 1) to these factors, solving for (y). This yields:

2y 1 1 rarr; y 0 (not a positive integer) 2y 1 5 rarr; y 2 2y 1 11 rarr; y 5 2y 1 55 rarr; y 27 2y 1 73 rarr; y 36 2y 1 365 rarr; y 182 2y 1 803 rarr; y 401 2y 1 4015 rarr; y 2007 (not a positive integer when substituted in original equations)

Deriving Corresponding Values for (x) and (z)

For each value of (y), we can derive the corresponding value of (x) using Equation E:

For y 2, we get x 1821, 2y 1 5 rarr; x 401, z xy - 1 803 For y 5, we get x 1807, 2y 1 11 rarr; x 914, z xy - 1 914 For y 27, we get x 3641, 2y 1 55 rarr; x 1007, z xy - 1 1007

Substituting (x) and (y) back into the original equations, we can verify these triplets:

(401, 2, 803) (914, 5, 914) (1007, 27, 1007)

These triplets of positive integers satisfy the given system of equations, demonstrating a clear analytical approach.

Conclusion

The process of finding all triplets of positive integers ((x, y, z)) for the given system of equations involves a combination of algebraic manipulation and factorization. By systematically testing possible values and verifying solutions, we can ensure a comprehensive solution set.

Key Takeaways

1. Elimination of one variable to simplify the system of equations.

2. Using factorization techniques to find potential values for the remaining variables.

3. Systematic verification of solutions to ensure correctness.