To find the expression that is equivalent to the given complex fraction
[tex]\[
\frac{\frac{-2}{x} + \frac{5}{y}}{\frac{3}{y} - \frac{2}{x}}
\][/tex]
let's simplify it step by step.
1. Identifying the Numerator and Denominator:
The given complex fraction can be split into:
[tex]\[
\text{Numerator} = \frac{-2}{x} + \frac{5}{y}
\][/tex]
[tex]\[
\text{Denominator} = \frac{3}{y} - \frac{2}{x}
\][/tex]
2. Finding a Common Denominator for the Numerator and Denominator:
For simplification, we find a common denominator for each part:
[tex]\[
\text{Numerator: } \frac{-2}{x} + \frac{5}{y} \quad \text{(common denominator = xy)}
\][/tex]
[tex]\[
= \frac{-2y + 5x}{xy}
\][/tex]
Similarly:
[tex]\[
\text{Denominator: } \frac{3}{y} - \frac{2}{x} \quad \text{(common denominator = xy)}
\][/tex]
[tex]\[
= \frac{3x - 2y}{xy}
\][/tex]
3. Combining the Numerator and Denominator:
Now, substitute these back into the original complex fraction:
[tex]\[
\frac{\frac{-2y + 5x}{xy}}{\frac{3x - 2y}{xy}}
\][/tex]
4. Simplifying the Complex Fraction:
To simplify, multiply by the reciprocal of the denominator:
[tex]\[
= \frac{-2y + 5x}{xy} \times \frac{xy}{3x - 2y}
\][/tex]
The \(\frac{xy}{xy}\) cancels out, leaving:
[tex]\[
= \frac{-2y + 5x}{3x - 2y}
\][/tex]
Thus, the simplified form of the given complex fraction is:
[tex]\[
\boxed{\frac{-2y + 5x}{3x - 2y}}
\][/tex]
