Solving the Equation x2y3 3xy: Methods and Insights

Have you ever encountered a problem that seems straightforward, only to find a surprising twist in its solution? Such is the case with the equation x2y3 3xy. This equation might appear complex at first glance, but with a methodical approach, we can uncover the solutions hidden within it. Let's delve into the various methods and insights that will guide us to the solution.

Introduction to the Equation

The equation in question is x2y3 3xy. This is an algebraic equation involving two variables, x and y. To find the solutions to this equation, we need to manipulate it algebraically and explore the possibilities that arise.

Initial Observations

At first glance, this equation seems intimidating. But let's break it down into smaller, more manageable parts. One of the first things we observe is that the equation can be simplified to reveal a clearer path to the solution.

Simplifying the Equation

Starting with the given equation:

x2y3 3xy

We can simplify it by moving all terms to one side of the equation:

x2y3 - 3xy 0

Next, we factor out the common term, which is xy:

xy(x2y2 - 3) 0

Factorization and Solving for x and y

From the factored form, we can see that the equation will be true if either of the factors equals zero:

Case 1: xy 0

If xy 0, this implies that either x 0 or y 0. These are the trivial solutions:

x 0 y 0

This is a straightforward solution, and it confirms that when either x or y is zero, the equation holds true.

Case 2: x2y2 - 3 0

For the equation to hold true in this case, we need:

x2y2 3

Dividing both sides by y2 (assuming y ≠ 0), we get:

x2 3/y2

Taking the square root of both sides, we get:

x ±√(3/y2) ±√(3)/y

General Solutions and Infinite Solutions in a Circle

From the above derived general solution, we can express x in terms of y as:

x ±√(3)/y

This equation represents a family of solutions. If we rewrite the solution in a geometric context, it describes a set of points that lie on a circle in the coordinate plane. Specifically, the equation can be rewritten as:

x2 y2 3

This is the equation of a circle centered at the origin with radius √3.

Conclusion

In summary, the equation x2y3 3xy has both trivial and non-trivial solutions. The trivial solutions are when either x or y is zero, while the non-trivial solutions form a circle in the coordinate plane. This example illustrates the importance of algebraic manipulation and the role of geometric interpretations in solving algebraic equations. Understanding these methods not only helps in solving complex equations but also provides deeper insights into the nature of mathematical relationships.

Keywords

Equation solving Algebraic manipulation Mathematical methods