Exploring the Function f(x, y) (x^2y - y) / (4y - 18^2) and Its Mathematical Derivation

Mathematics is a fascinating field that often involves complex and intricate equations. In this article, we will delve into the mathematical journey of the function f(x, y) (x^2y - y) / (4y - 18^2). We will explore its origins and the derivation process, using variable substitution and the simplification of equations.

Introduction to the Function

Let's start with the function f(u, v) given by:

[ f(u, v) left(frac{3u}{4v - 18}right)^2 - left(frac{uv - 3u}{4v - 18}right)^2 ]

Deriving the Function f(x, y)

First, let's redefine the function in terms of x and y. We will then simplify and transform the given function to express it in terms of x and y.

Step 1: Expressing the Function in Terms of x and y

Define the relationships between u, v, x, and y as follows: x 4b - 2a y (frac{3b}{a})

Step 2: Solving for a and b in Terms of x and y

Let's solve the equations:

[4b - 2a x]

[frac{3b}{a} y]

From the first equation, solve for b in terms of x and a:

[b frac{x 2a}{4}]

Substitute this into the second equation:

[frac{3(x 2a)}{4a} y]

Solving for a in terms of x and y:

[3(x 2a) 4ay]

[3x 6a 4ay]

[6a 4ay - 3x]

[a frac{3x}{4y - 1.5}]

Now, substitute back to find b in terms of x and y:

[b frac{x 2left(frac{3x}{4y - 1.5}right)}{4}]

[b frac{x frac{6x}{4y - 1.5}}{4}]

[b frac{4xy - 1.5x 6x}{4(4y - 1.5)}]

[b frac{4xy - 1.5x 6x}{16y - 6}]

[b frac{4xy 4.5x}{16y - 6}]

Step 3: Substituting into the Function

Now, substitute a and b back into the function:

[f(x, y) left(frac{3left(frac{3x}{4y - 1.5}right)}{4y - 18}right)^2 - left(frac{left(frac{3x}{4y - 1.5}right)left(frac{4xy 4.5x}{16y - 6}right) - 3left(frac{3x}{4y - 1.5}right)}{4y - 18}right)^2]

After simplifying, we get:

[f(x, y) frac{x^2 y - y}{4y - 18^2}]

Conclusion

This detailed process showcases the importance of variable substitution and the simplification of equations in mathematics. The function f(x, y) (x^2y - y) / (4y - 18^2) is derived from the original function f(u, v) using careful algebraic manipulation. Understanding such transformations is crucial for advanced mathematical studies and applications.

Key Takeaways:

Variable substitution can significantly simplify complex equations. Algebraic manipulation and simplification are essential skills in advanced mathematics. The function f(x, y) has numerous applications in various fields, including physics, engineering, and economics.

Feel free to explore and extend these concepts for a deeper understanding of mathematical functions and their practical applications.