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Which expression is equivalent to the following complex fraction?

[tex]\[
\frac{\frac{-2}{x}+\frac{5}{y}}{\frac{3}{y}-\frac{2}{x}}
\][/tex]

A. \(\frac{-2 y + 5 x}{3 x - 2 y}\)

B. \(\frac{3 x - 2 y}{-2 y + 5 x}\)

C. \(\frac{x^2 y^2}{(-2 y + 5 x)(3 x - 2 y)}\)

D. [tex]\(\frac{(-2 y + 5 x)(3 x - 2 y)}{x^2 y^2}\)[/tex]



Answer :

GinnyAnswer
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]

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