Question

which expression is equivalent to the following complex fraction?
\\(\frac{\frac{2}{x} - \frac{4}{y}}{\frac{-5}{y} + \frac{3}{x}}\\)
options:
\\(\frac{3y + 5x}{2(y - 2x)}\\)
\\(\frac{2(y - 2x)}{3y - 5x}\\)
\\(\frac{2(y - 2x)(3y - 5x)}{x^2 y^2}\\)
\\(\frac{x^2 y^2}{2(y - 2x)(3y - 5x)}\\)

Explanation:

Step1: Simplify numerator

The numerator is $\frac{2}{x} - \frac{4}{y}$. Find a common denominator, which is $xy$. So we have $\frac{2y - 4x}{xy}=\frac{2(y - 2x)}{xy}$.

Step2: Simplify denominator

The denominator is $\frac{-5}{y}+\frac{3}{x}$. The common denominator is $xy$, so it becomes $\frac{-5x + 3y}{xy}=\frac{3y - 5x}{xy}$.

Step3: Divide numerator by denominator

Now we have $\frac{\frac{2(y - 2x)}{xy}}{\frac{3y - 5x}{xy}}$. Dividing by a fraction is multiplying by its reciprocal, so $\frac{2(y - 2x)}{xy}\times\frac{xy}{3y - 5x}$. The $xy$ terms cancel out, leaving $\frac{2(y - 2x)}{3y - 5x}$.

Answer:

$\frac{2(y - 2x)}{3y - 5x}$ (corresponding to the second option)