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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]


Sagot :

To solve and simplify the given complex fraction:

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

we need to follow these steps:

### Step 1: Find a common denominator for the individual fractions in both the numerator and the denominator.

For the numerator \(\frac{-2}{x} + \frac{5}{y}\):
- The common denominator is \(xy\), so:
[tex]\[ \frac{-2}{x} = \frac{-2y}{xy} \quad \text{and} \quad \frac{5}{y} = \frac{5x}{xy} \][/tex]
Combining these fractions:
[tex]\[ \frac{-2}{x} + \frac{5}{y} = \frac{-2y + 5x}{xy} \][/tex]

For the denominator \(\frac{3}{y} - \frac{2}{x}\):
- The common denominator is \(xy\), so:
[tex]\[ \frac{3}{y} = \frac{3x}{xy} \quad \text{and} \quad \frac{2}{x} = \frac{2y}{xy} \][/tex]
Combining these fractions:
[tex]\[ \frac{3}{y} - \frac{2}{x} = \frac{3x - 2y}{xy} \][/tex]

### Step 2: Substitute these expressions back into the original complex fraction.

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

### Step 3: Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator.

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

### Step 4: Simplify and look at the resulting fraction to match it with one of the given choices.

The resulting fraction is:

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

### Conclusion:
The equivalent expression to the given complex fraction is:

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

Therefore, the correct answer is:

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