To determine which expression is equivalent to the given complex fraction:
[tex]\[
\frac{\frac{-2}{x} + \frac{5}{y}}{\frac{3}{y} - \frac{2}{x}}
\][/tex]
we need to simplify this complex fraction step-by-step.
1. Combine the numerators and denominators:
- For the numerator:
[tex]\[
\frac{-2}{x} + \frac{5}{y}
\][/tex]
To combine these fractions, find a common denominator (which is [tex]\(xy\)[/tex]):
[tex]\[
\frac{-2y}{xy} + \frac{5x}{xy} = \frac{-2y + 5x}{xy}
\][/tex]
- For the denominator:
[tex]\[
\frac{3}{y} - \frac{2}{x}
\][/tex]
Again, find a common denominator (which is [tex]\(xy\)[/tex]):
[tex]\[
\frac{3x}{xy} - \frac{2y}{xy} = \frac{3x - 2y}{xy}
\][/tex]
2. Write the complex fraction using the combined numerators and denominators:
[tex]\[
\frac{\frac{-2y + 5x}{xy}}{\frac{3x - 2y}{xy}}
\][/tex]
3. Divide the numerators and denominators:
When dividing fractions, multiply by the reciprocal of the divisor:
[tex]\[
\frac{-2y + 5x}{xy} \div \frac{3x - 2y}{xy} = \frac{-2y + 5x}{xy} \cdot \frac{xy}{3x - 2y} = \frac{(-2y + 5x) \cdot xy}{xy \cdot (3x - 2y)}
\][/tex]
4. Simplify the expression:
The [tex]\(xy\)[/tex] terms cancel out in the numerator and denominator:
[tex]\[
\frac{-2y + 5x}{3x - 2y}
\][/tex]
Therefore, the simplified expression equivalent to the given complex fraction is:
[tex]\[
\frac{-2 y + 5 x}{3 x - 2 y}
\][/tex]
Comparing this with the given choices, the equivalent expression is:
[tex]\[
\boxed{\frac{-2 y + 5 x}{3 x - 2 y}}
\][/tex]
