Exploring Ordered Pairs for the Equation x2y -3

When tackling the equation x2y -3, we delve into a fascinating exploration of its ordered pairs, denoting solutions that satisfy this relationship. The problem is inherently nonlinear, involving a quadratic term in x, and leads us to discover an infinite set of solutions. Each ordered pair (x, y) represents a valid solution to the equation, and this article will guide you through the process of identifying and understanding these points.

Understanding the Equation x2y -3

The equation x2y -3 is fundamentally a nonlinear equation due to the presence of the term x2. For a given x value, y must be calculated such that their product equals -3. This equation does not represent a straight line, despite the possibility of infinite solutions, and instead showcases a curve or a parabola in the coordinate system. The transformation of this equation to the form y -frac{x3}{2} reveals the underlying relationship between x and y more clearly.

Identifying Ordered Pairs

Let's start by identifying some obvious solutions to the equation. We know that for x -1, the value of y must satisfy (-1)2y -3, leading to y -3. Therefore, one solution is the ordered pair (-1, -3). Similarly, when x 1, we have (1)2y -3, which simplifies to y -3, giving us the ordered pair (1, -3). These are just a couple of the innumerable solutions that lie on the curve defined by the equation.

Deriving General Solutions

For a general value of x, let's denote it as t. Substituting t for x in the equation, we get:

t2y -3

Solving for y, we have:

y -frac{3}{t2}

This shows that any real number t will yield a corresponding y value, thus confirming the existence of an infinite set of ordered pairs. The relationship y -frac{3}{x2} illustrates that as x increases or decreases in magnitude, the absolute value of y decreases as the denominator of the fraction grows, keeping the product at -3.

Graphical Representation and Solutions

To further visualize the solutions, we can plot several ordered pairs derived from the general formula. For example:

When t -3, y -frac{3}{(-3)2} -frac{3}{9} -frac{1}{3}. The ordered pair is (-3, -frac{1}{3}). With t 3, y -frac{3}{32} -frac{3}{9} -frac{1}{3}. The ordered pair is (3, -frac{1}{3}). For t -sqrt{2}, y -frac{3}{(-sqrt{2})2} -frac{3}{2}. The ordered pair is (-sqrt{2}, -frac{3}{2}). When t sqrt{2}, y -frac{3}{(sqrt{2})2} -frac{3}{2}. The ordered pair is (sqrt{2}, -frac{3}{2}).

Conclusion

In conclusion, the equation x2y -3 encapsulates an infinite set of ordered pairs, demonstrating the rich and complex nature of nonlinear relationships. Understanding these solutions not only deepens our mathematical intuition but also provides a valuable resource for various applications in mathematics and beyond. By exploring the properties of this equation, we gain insights into its graphical representation and the behavior of its solutions, enhancing our ability to analyze and solve related problems effectively.