How to Find All Real Number Triples Satisfying the Given Equations

When solving mathematical problems involving real number triples, it's crucial to understand the underlying equations and the methods used to find their solutions. In this article, we will explore a detailed process for finding all real number triples that satisfy a specific set of equations, namely:

xy z2 {E01} yz x2 {E02} zx y2 {E03}

Let's begin by understanding the given equations and the methodology used to find the solutions.

Equation Analysis

The equations {E01}, {E02}, and {E03} establish a relationship between the variables x, y, and z. To simplify the problem, we start by subtracting one equation from another to eliminate variables and derive new equations.

Subtraction and Simplification

Subtracting {E02} from {E01} and {E03} from {E02} gives:

E04: x - z · xyz 1 E05: y - x · xyz 1

From {E04} and {E05}, we can derive:

E06: 2x yz 4x2 yz2 - 2yz y2 y - z2 8

Let's solve {E06} and {E07} simultaneously:

E07: y x ± √2, z x ± √2

Application of Solutions

Substitute the solutions from {E07} into {E01} to find the values of x:

x · (x ± √2) - 1 x ± √2

This simplifies to:

1 - x2 ± √2 · x x2 ± 2 · x ± 2√2 · x

4√2 · x ± 1

Therefore:

x ± frac{1}{3√2}

Substituting these values into the expressions for y and z gives:

y x ± √2 ± frac{1}{3√2} ± √2

z x ± √2 ± frac{1}{3√2} ± √2

The final solutions are:

xyz {± frac{1}{3√2}, ± frac{7}{3√2}, ± frac{5}{3√2}}

Conclusion

In this article, we have explored the process of solving a system of equations to find all possible real number triples. The methodology involves strategic manipulation of the given equations and substitution of derived relations. Understanding such methods is essential for solving similar problems in algebra and mathematical analysis.

For more information on real number triples and numerical methods, refer to the resources and further reading sections below.