Alright, let's dissect and solve the given complex fraction step-by-step. We need to simplify the following expression:
[tex]\[
\frac{\frac{-2}{x}+\frac{5}{y}}{\frac{3}{y}-\frac{2}{x}}
\][/tex]
### Step 1: Combine Fractions in Numerator and Denominator
First, we want to combine the fractions in both the numerator and the denominator:
#### Numerator:
[tex]\[
\frac{-2}{x} + \frac{5}{y}
\][/tex]
To combine these, we get a common denominator, which is [tex]\( x \times y \)[/tex]:
[tex]\[
\frac{-2y}{xy} + \frac{5x}{xy} = \frac{-2y + 5x}{xy}
\][/tex]
#### Denominator:
[tex]\[
\frac{3}{y} - \frac{2}{x}
\][/tex]
Similarly, combining these with a common denominator of [tex]\( y \times x \)[/tex] yields:
[tex]\[
\frac{3x}{xy} - \frac{2y}{xy} = \frac{3x - 2y}{xy}
\][/tex]
### Step 2: Simplify the Complex Fraction
Now we have:
[tex]\[
\frac{\frac{-2y + 5x}{xy}}{\frac{3x - 2y}{xy}}
\][/tex]
To simplify this, we can divide the numerator by the denominator:
[tex]\[
\frac{\frac{-2y + 5x}{xy}}{\frac{3x - 2y}{xy}} = \frac{-2y + 5x}{3x - 2y}
\][/tex]
### Step 3: Check the Expressions
From the simplification process, we see that our result is:
[tex]\[
\frac{-2y + 5x}{3x - 2y}
\][/tex]
Comparing this with the given options:
[tex]\[
\frac{-2 y+5 x}{3 x-2 y}
\][/tex]
[tex]\[
\frac{3 x-2 y}{-2 y+5 x}
\][/tex]
[tex]\[
\frac{x^2 y^2}{(-2 y+5 x)(3 x-2 y)}
\][/tex]
[tex]\[
\frac{(-2 y+5 x)(3 x-2 y)}{x^2 y^2}
\][/tex]
We unequivocally see that:
[tex]\[
\frac{-2 y+5 x}{3 x-2 y}
\][/tex]
matches our simplified expression. Therefore, the expression equivalent to the given complex fraction is:
[tex]\[
\boxed{\frac{-2 y+5 x}{3 x-2 y}}
\][/tex]
