Question

simultaneous equation. solve.

Answer 1

$\underline{ \bold{ \red{SOLUTION \: \: \: }}}$➡

$\frac{x + y}{xy} = 2 \: \: - - - - - > (1)\\ \\ \frac{x - y}{xy} = 6 \: \: - - - - - > (2) \\ \\ (1) = > xy = \frac{x + y}{2} \: \: - - - - > (3) \\ \\ (2) = > xy = \frac{x - y}{6}\: \: - - - - - > (4) \\ \\ \underline{ \: \: From \: \: (3) \: \: and \: \: (4) \: \: } \\ \\ \frac{x + y}{2} = \frac{x - y}{6} \\ \\ = > x + y = \frac{x - y}{3} \\ \\ = > 3x + 3y = x - y \\ \\ = > 2x + 4y = 0 \\ \\ = > 2(x + 2y) = 0 \\ \\ = > x + 2y = 0 \\ \\ = > x = - 2y \: \: - - - - > (5) \\ \\ (1) = > \frac{x + y}{xy} = 2 \\ \\ = > \frac{( - 2y) + y}{( - 2y )\times y} = 2 \\ \\ = > \frac{ - y}{ - 2y {}^{2} } = 2 \\ \\ = > \frac{1}{2y} = 2 \\ \\ = > 4y = 1 \\ \\ = > y = \frac{1}{4} \\ \\ (5) = > x = - 2y \\ \\ = > x = - 2 \times ( \frac{1}{4} ) \\ \\ = > x = - \frac{1}{2}$

$\underline{ \bold{ \red{ANSWER\: \: }}}$ ➡ $\boxed{x = - \: \frac{1}{2} \: \: \: \: ,\: \: \: y = \frac{1}{4} }$

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⭐$\bold{\red{.THANKS.}}$⭐

Answer 2

$\underline{\underline{ \bold{ \red{SOLUTION \: \: \: }}}}$➡

$\frac{x + y}{xy} = 2 \: \: - - - - - > (1)\\ \\ \frac{x - y}{xy} = 6 \: \: - - - - - > (2) \\ \\ (1) = > xy = \frac{x + y}{2} \: \: - - - - > (3) \\ \\ (2) = > xy = \frac{x - y}{6}\: \: - - - - - > (4) \\ \\ \underline{ \: \: From \: \: (3) \: \: and \: \: (4) \: \: } \\ \\ \frac{x + y}{2} = \frac{x - y}{6} \\ \\ = > x + y = \frac{x - y}{3} \\ \\ = > 3x + 3y = x - y \\ \\ = > 2x + 4y = 0 \\ \\ = > 2(x + 2y) = 0 \\ \\ = > x + 2y = 0 \\ \\ = > x = - 2y \: \: - - - - > (5) \\ \\ (1) = > \frac{x + y}{xy} = 2 \\ \\ = > \frac{( - 2y) + y}{( - 2y )\times y} = 2 \\ \\ = > \frac{ - y}{ - 2y {}^{2} } = 2 \\ \\ = > \frac{1}{2y} = 2 \\ \\ = > 4y = 1 \\ \\ = > y = \boxed{ \frac{1}{4}} \\ \\ (5) = > x = - 2y \\ \\ = > x = - 2 \times ( \frac{1}{4} ) \\ \\ = > x = \boxed{- \frac{1}{2}}$

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$\underline{ \underline{\bold{ \red{VERIFICATION \: \: \: }}}}$ ➡

$\ \underline{\bold{From \: \: eq.( 1) \: \: }} \\ \\ \bold{ L.H.S.= \frac{x + y}{xy} } \\ \\ \bold{ = \frac{ ( - \frac{1}{2} ) + \frac{1}{4} }{( - \frac{1}{2} ) \times \frac{1}{4} } } \\ \\ \bold{ = \frac{ \frac{ -2 + 1}{4} }{ - \frac{1}{8} } } \\ \\ \bold { = \frac{ - \frac{ 1}{4} }{ - \frac{1}{8} } } \\ \\ \bold{ = (- \frac{1}{4}) \times ( - \frac{8}{1}) } \\ \\ = \bold{2} \\ \\ \bold{ = R.H.S.}$

$\underline{\bold{ \: From \: \: eq.(2) \: \: }} \\ \\\bold{ L.H.S.= \frac{x - y}{xy} } \\ \\ \bold{ = \frac{ ( - \frac{1}{2} ) - \frac{1}{4} }{( - \frac{1}{2} ) \times \frac{1}{4} } } \\ \\ \bold{ = \frac{ \frac{ -2 - 1}{4} }{ - \frac{1}{8} } } \\ \\ \bold { = \frac{ - \frac{ 3}{4} }{ - \frac{1}{8} } } \\ \\ \bold{ = (- \frac{3}{4}) \times ( - \frac{8}{1}) } \\ \\ = \bold{3 \times 2} \\ \\ \bold{ = 6} \: \: \\ \\ \bold{ = R.H.S.}$

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$\underline{ \bold{ \red{ANSWER\: \: }}}$ ➡ $\boxed{x = - \: \frac{1}{2} \: \: \: \: ,\: \: \: y = \frac{1}{4} }$

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✝$\bold{\red{THANKS...}}$✝