Solving Real Number Equations: A Comprehensive Guide

In this article, we will explore the process of solving real number equations through a step-by-step approach. The problem presented involves a series of equations that require manipulation and reasoning to find the values of variables x, y, and z. By understanding and applying the principles of algebra, we can arrive at a solution.

Problem Statement

We are given the following conditions:

[ frac{xyz}{zx} 2 ]

[ frac{xyz}{yz} 4 ]

Our goal is to find the value of [ frac{xyz}{xy} ].

Step-by-Step Solution

Let's start by simplifying the given equations.

xyzzx2xzxyz12 xyzyz4yzxyz14

From the first equation, we can write:

[ frac{xyz}{zx} 2 iff frac{1}{frac{zx}{xyz}} 2 iff frac{1}{frac{1}{xy}} 2 iff frac{xy}{yz} 2 ]

Similarly, from the second equation, we have:

[ frac{xyz}{yz} 4 iff frac{1}{frac{yz}{xyz}} 4 iff frac{1}{frac{1}{xy}} 4 iff frac{xy}{zx} 4 ]

We denote:

[ p frac{xyz}{xy} Rightarrow frac{xy}{xyz} frac{1}{p} ]

Adding the Equations

Now, we add the simplified equations:

[ 2 frac{1}{2} frac{1}{4} frac{1}{p} ]

Rewriting and solving for [ p ]:

[ frac{1}{p} 2 - frac{3}{4} frac{5}{4} Rightarrow p frac{4}{5} ]

Conclusion

The value of [ frac{xyz}{xy} ] is [ frac{4}{5} ]. This problem demonstrates the use of algebraic manipulation to solve a system of equations involving real numbers. Understanding these techniques can be applied in various mathematical contexts, including real-world problem solving.

This article is structured to guide both beginners and intermediate learners in solving real number equations effectively. Whether you are preparing for a math contest or simply looking to improve your algebraic problem-solving skills, the method presented here can be a valuable resource.