Solving for x, y, and z in Equations: A Comprehensive Guide

Algebra is a fundamental aspect of mathematics, offering a myriad of applications in various fields. This article seeks to solve a specific set of equations involving variables x, y, and z. By understanding the systematic approach to solving simultaneous equations, we can find the values of x, y, and z. Let's explore this problem in depth.

Given the following equations:

xy -1 yz 1 zx 0

Step-by-Step Solution

We start by analyzing the given equations and attempting to find the values of x, y, and z.

1. Direct Substitution and Observation

Let's begin by substituting the third equation into the other two.

From the third equation, we have: zx 0. Since z is a multiplier, if z is not zero, then x must be zero. However, we will explore other possibilities. Using the third equation, we can see that if z is zero, then the second equation becomes yz 1, which is impossible unless y is undefined or z is non-zero. Therefore, z must be zero.

If z 0, then from the second equation:

yz 1 implies 0 * y 1 which is not possible. Therefore, z cannot be zero. However, the only logical approach here is to use the given equations.

2. Systematic Approach

A more structured approach involves combining the equations:

From xy -1 and yz 1, we can express y in terms of x and z.

Substituting z -x:

From the third equation: zx 0 implies x(-x) 0 or -x^2 0. Therefore, x must be 0 or -1.

Let's test these values:

If x 0: xy -1 implies 0 * y -1, which is not possible. If x -1: xy -1 implies -1 * y -1, so y 1. yz 1 implies 1 * z 1, so z 1. Finally, checking the third equation: zx 0 implies -1 * 1 -1, which is consistent.

Hence, the solution is:

x -1 y 0 z 1

3. Generalization of the Problem

Given the general form of equations:

xy a yz b zx c

We can deduce the following:

2xyz abc xyz frac{abc}{2}

Multiplying the equation xyz frac{abc}{2} by each variable and then subtracting the corresponding terms:

x xyz - yz frac{abc}{2} - b frac{a - 2b}{2} y xyz - zx frac{abc}{2} - c frac{ab - 2c}{2} z xyz - xy frac{abc}{2} - a frac{bc - 2a}{2}

Substituting a -1, b 1, and c 0:

x frac{-1 - 2(1)}{2} -frac{3}{2} y frac{(-1)(1) - 0}{2} -frac{1}{2} z frac{(1)(0) - 2(-1)}{2} frac{2}{2} 1

Therefore, the values are:

x -1 y 0 z 1

Conclusion

In summary, the values of x, y, and z that satisfy the given equations are (boxed{x -1}), (boxed{y 0}), and (boxed{z 1}). This solution demonstrates the systematic approach to solving simultaneous algebraic equations and can be applied to a wider range of similar problems.