In the equations $3 x + 2 y = 13 x y$ and $4 x - 5 y = 2 x y$ , the values of x and y that satisfy the equations are

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$(\frac{1}{2}, \frac{1}{3})$

$(\frac{1}{3}, \frac{1}{2})$

(2, 3 )

(3, 2)

answer is C.

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Detailed Solution

$3 x + 2 y = 13 x y . . . (i) \times 5 4 x - 5 y = 2 x y . . . (i i) \times 2__________________\begin{array}{l} 15 x + 10 y = 65 x y \\ 8 x - 10 y = 4 x y \end{array}________________23 x = 69 x y$

$\begin{array}{l} \Rightarrow 23 x = 69 x y \Rightarrow x = 3 x y \\ \Rightarrow 1 = 3 y \Rightarrow y = \frac{1}{3} \end{array}$

In equation (i) $\to (y = \frac{1}{3})$

$\begin{array}{l} 3 x + 2 y = 13 x y \\ 3 x + 2 \times \frac{1}{3} = 13 x \times \frac{1}{3} \\ 3 x + \frac{2}{3} = \frac{13 x}{3} \\ \frac{2}{3} = \frac{13 x}{3} - 3 x = \frac{13 x - 9 x}{3} = \frac{4 x}{3} \\ \Rightarrow \frac{2}{3} = \frac{4 x}{3} \\ \Rightarrow 4 x = 2 \end{array}$