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Solve for x and y

i) 2a/x +3b/y +1 = 0, 3a/x-b/y - 4 =0

ii) 2xy/x+y = 3/2 ; xy/2x - y =-3/10 ,x+y is not equal to 0, 2x - y is not equal to zero

$(i) Given \frac{2 a}{x} + \frac{3 b}{y} + 1 = 0 - - - (i) \frac{3 a}{x} - \frac{b}{y} - 4 = 0 - - - (i i) Let \frac{1}{x} = u and \frac{1}{y} = v Then the equation becomes 2 a u + 3 b v + 1 = 0 - - - (i i i) 3 a u - b v - 4 = 0 - - - - (i v) Now multiply eq(iv) with 3 2 a u + 3 b v + 1 = 0 9 a u - 3 b v - 12 = 0 By solving we get u = \frac{1}{a} v = \frac{- 1}{v} \Rightarrow x = a and y=-b (i i) \frac{2 x y}{x + y} = \frac{3}{2} \Rightarrow \frac{x + y}{2 x y} = \frac{2}{3} \Rightarrow \frac{1}{2 y} + \frac{1}{2 x} = \frac{2}{3} \Rightarrow \frac{1}{x} + \frac{1}{y} = \frac{4}{3} \frac{x y}{2 x - y} = \frac{- 3}{10} \Rightarrow \frac{2 x - y}{x y} = \frac{- 10}{3} \Rightarrow \frac{2}{y} - \frac{1}{x} = \frac{- 10}{3} Let \frac{1}{x} = u and \frac{1}{y} = v Then the equation becomes v+u= \frac{4}{3} 2 v - u = \frac{- 10}{3} By solving we get v= \frac{- 2}{3} u = 2 \Rightarrow x = \frac{1}{2} y = \frac{- 3}{2}$

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