Question

Solve the following simultaneous equation.

\[ \frac{2}{x} - \frac{3}{y} = 15; \frac{8}{x} + \frac{5}{y} = 77\]

Answer 1

\[\frac{2}{x} - \frac{3}{y} = 15; \frac{8}{x} + \frac{5}{y} = 77\]

\[\text{ Let }\frac{1}{x} = u\text{ and }\frac{1}{y} = v\]

So, the equation becomes

\[2u - 3v = 15 . . . . . \left( I \right)\]

\[8u + 5v = 77 . . . . . \left( II \right)\]

Multiply (I) with 4 we get

\[8u - 12v = 60 . . . . . \left( III \right)\]

(II) − (III)

\[8u - 8u + 5v - \left( - 12v \right) = 77 - 60\]

\[ \Rightarrow 17v = 17\]

\[ \Rightarrow v = 1\]

\[\text{ Putting the value of v in }\left( I \right)\]

\[2u - 3\left( 1 \right) = 15\]

\[ \Rightarrow 2u = 15 + 3 = 18\]

\[ \Rightarrow u = 9\]

Thus, 

\[\frac{1}{x} = u = 9\]

\[ \Rightarrow x = \frac{1}{9}\]

\[\frac{1}{y} = v = 1\]

\[ \Rightarrow y = 1\]

\[\left( x, y \right) = \left( \frac{1}{9}, 1 \right)\]