Let's break down the given complex fraction and simplify it step by step.
Given the complex fraction:
[tex]\[
\frac{\frac{2}{x} - \frac{4}{y}}{\frac{-5}{y} + \frac{3}{x}}
\][/tex]
First, combine the terms in the numerator and denominator.
The common denominator for the fractions in the numerator is [tex]\(xy\)[/tex]:
[tex]\[
\frac{2}{x} - \frac{4}{y} = \frac{2y}{xy} - \frac{4x}{xy} = \frac{2y - 4x}{xy}
\][/tex]
Similarly, the common denominator for the fractions in the denominator is [tex]\(xy\)[/tex]:
[tex]\[
\frac{-5}{y} + \frac{3}{x} = \frac{-5x}{xy} + \frac{3y}{xy} = \frac{-5x + 3y}{xy}
\][/tex]
So the complex fraction becomes:
[tex]\[
\frac{\frac{2y - 4x}{xy}}{\frac{-5x + 3y}{xy}}
\][/tex]
To simplify, multiply by the reciprocal of the denominator:
[tex]\[
\frac{2y - 4x}{xy} \div \frac{-5x + 3y}{xy} = \frac{2y - 4x}{xy} \cdot \frac{xy}{-5x + 3y}
\][/tex]
The [tex]\(xy\)[/tex] terms cancel out:
[tex]\[
\frac{2y - 4x}{-5x + 3y}
\][/tex]
Now we need to find which of the given expressions is equivalent to this simplified form:
[tex]\[
\frac{2y - 4x}{-5x + 3y}
\][/tex]
Let's look at each given expression one by one:
1. [tex]\(\frac{3y + 5x}{2(y - 2x)}\)[/tex]
2. [tex]\(\frac{2(y - 2x)}{3y - 5x}\)[/tex]
[tex]\[
\text{This can be rewritten as } \frac{2y - 4x}{3y - 5x}
\][/tex]
3. [tex]\(\frac{2(y - 2x)(3y - 5x)}{x^2 y^2}\)[/tex]
4. [tex]\(\frac{x^2 y^2}{2(y - 2x)(3y - 5x)}\)[/tex]
Comparing the simplified form [tex]\(\frac{2y - 4x}{-5x + 3y}\)[/tex] with each option, we see that expression 2 is equivalent when you consider that:
[tex]\[
\frac{2(y - 2x)}{3y - 5x} = \frac{2y - 4x}{3y - 5x}
\][/tex]
Thus, the equivalent expression is:
[tex]\(\boxed{\frac{2(y - 2x)}{3y - 5x}}\)[/tex]
