Correct option is (A) x = 3, y = 2

\(\frac{10}{x+y}+\frac{2}{x-y}= 4\)    ________(1)

and \(\frac{15}{x+y}-\frac{5}{x-y}=-2\)   ________(2)

Put \(\frac1{x+y}=X\;\&\;\frac{1}{x-y}=Y\)

Then equations (1) & (2) converts to

10X + 2Y = 4           ________(3)

and 15X - 5Y = -2   ________(4)

Multiply equation (3) by \(\frac52,\) we get

25X + 5Y = 10        ________(5)

By adding equation (4) & (5), we get

(25X + 5Y) + (15X - 5Y) = 10 - 2

\(\Rightarrow\) 40X = 8

\(\Rightarrow\) X \(=\frac8{40}=\frac15\)

Then from (3), we get

\(10\times\frac15+2Y=4\)

\(\Rightarrow\) 2 + 2Y = 4

\(\Rightarrow\) 2Y = 4 - 2 = 2

\(\Rightarrow\) Y = \(\frac22\) = 1

\(\because\) \(X=\frac15\)

\(\Rightarrow\) \(\frac1{x+y}=\frac{1}5\)

\(\Rightarrow\) x+y = 5        ________(6)

and Y = 1

\(\Rightarrow\) \(\frac1{x-y}=1\)

\(\Rightarrow\) x - y = 1        ________(7)

By adding equations (6) & (7), we get

2x = 5+1 = 6

\(\Rightarrow\) x = \(\frac62\) = 3

\(\therefore\) y = 5 - x

= 5 - 3 = 2

Correct option is A) x = 3, y = 2