Solving the Equation xy xy x/y: A Step-by-Step Analysis

When dealing with the equation xy xy x/y, it is important to understand the constraints and properties of the variables involved to solve the problem correctly. This document walks through the detailed process of solving this equation, ensuring clarity and thoroughness in each step.

Understanding the Equation

Given the equation:

xy xy xy x/y

The first step is to recognize that the equation is consistent and does not introduce any new information beyond the second equation. Let's start by analyzing the constraint that y ≠ 0 because division by zero is undefined in mathematics.

Solving the Equation

Since we know that y ≠ 0, let's begin by manipulating the equation xy x/y.

Step 1: Basic Manipulations

Multiply both sides of the equation xy x/y by y. This gives us: xy2 x Rearrange the terms to set the equation to zero: xy2 - x 0 Factor out x: x(y2 - 1) 0

This factored form gives us two potential solutions:

x 0 y2 - 1 0

Let's analyze these solutions one by one.

Step 2: Testing the Solutions

Solution 1: x 0

Using the original equation xy x/y, substitute x 0: 0 * y x / y becomes 0 * y 0 / y (which simplifies to 0 0) Although this simplification is true, it does not provide a meaningful solution for y. Therefore, x 0 and y ≠ 0 is not a valid solution for the problem.

Solution 2: y2 - 1 0

Solving the equation y2 - 1 0 for y: y2 1 y ±1 The possible values for y are y 1 and y -1.

CASE 1: y 1

Substitute y 1 into the original equation xy x/y: x * 1 x / 1 simplifies to x x This is always true, but does not provide a specific value for x except for x ≠ 0 as discussed earlier.

CASE 2: y -1

Substitute y -1 into the original equation xy x/y: x * -1 x / -1, which simplifies to -x -x Multiplying both sides by -1 gives us 2x 1 x 1/2

Therefore, we have determined that x 0.5 and y -1 satisfies the equation. Let's verify the solution.

Verification

xy 0.5 * -1 -0.5 xy 0.5 * -1 -0.5 x/y 0.5 / -1 -0.5

The solution checks out and solves the given equation correctly.

Conclusion

The value of x and y that satisfy the equation xy xy x/y is x 0.5 and y -1.