Answer :
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].
[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].
