To simplify the expression

$\frac{9x-4y}{x^{2}-y^{2}} - \frac{1}{2y-2x},$

we can first rewrite $x^2 - y^2$ as $(x - y)(x + y)$ and factor out the denominator of the second term:

1. The second term can be rewritten as $\frac{1}{2(y-x)} = -\frac{1}{2(x-y)}$.

Thus, the expression becomes:

$\frac{9x - 4y}{(x - y)(x + y)} + \frac{1}{2(x - y)}.$

Now, we need a common denominator to combine the two fractions. The common denominator will be $2(x - y)(x + y)$.

Now, we rewrite each term:

1. The first term: $\frac{9x - 4y}{(x - y)(x + y)} \cdot \frac{2}{2} = \frac{2(9x - 4y)}{2(x - y)(x + y)} = \frac{18x - 8y}{2(x - y)(x + y)}.$

2. The second term: $-\frac{1}{2(x - y)} \cdot \frac{(x + y)}{(x + y)} = -\frac{x + y}{2(x - y)(x + y)}.$

Now we can combine the two fractions:

$\frac{18x - 8y - (x + y)}{2(x - y)(x + y)} = \frac{18x - 8y - x - y}{2(x - y)(x + y)} = \frac{17x - 9y}{2(x - y)(x + y)}.$

Thus, the simplified expression is:

$\frac{17x - 9y}{2(x - y)(x + y)}.$

If you need further details or have any questions, feel free to ask!

Here are some related questions you might consider:

1. What happens to the expression if $y = 0$?
2. How do we determine the domain of the original expression?
3. Can you simplify the expression further under certain conditions?
4. What is the behavior of the expression as $x$ approaches $y$?
5. How does this expression change if we switch $x$ and $y$?

Tip: Always look for common factors when simplifying expressions; they can significantly reduce complexity.