Question

Solve the following simultaneous equations.

\[\frac{7x - 2y}{xy} = 5 ; \frac{8x + 7y}{xy} = 15\]

Answer 1

\[\frac{7x - 2y}{xy} = 5 ; \frac{8x + 7y}{xy} = 15\]

\[\Rightarrow \frac{7}{y} - \frac{2}{x} = 5\text{ and }\frac{8}{y} + \frac{7}{x} = 15\]
Let \[\frac{1}{x} = u, \frac{1}{y} = v\]
\[7v - 2u = 5 . . . . . \left( I \right)\]
\[8v + 7u = 15 . . . . . \left( II \right)\]
Multiply (I) with 7 and (II) with 2
\[49v - 14u = 35 . . . . . \left( III \right)\]
\[16v + 14u = 30 . . . . . \left( IV \right)\]
Adding (III) and (IV)
\[65v = 65\]
\[ \Rightarrow v = 1\]
\[\text{ And }\frac{1}{y} = v = 1\]
\[ \Rightarrow y = 1\]
Putting the value of v in (I)
\[7\left( 1 \right) - 2u = 5\]
\[ \Rightarrow u = 1\]
\[\frac{1}{x} = u = 1\]
\[ \Rightarrow x = 1\]
\[\left( x, y \right) = \left( 1, 1 \right)\]