If ${9 x}^{2} + {25 y}^{2} = 181$ and xy = - 6. Find the value of 3x + 5y

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Solution

${9 x}^{2} + {25 y}^{2} = 181$
$(3 x)^{2} + {(5 y)}^{2} = 181$
$\Rightarrow {(3 x)}^{2} + {(5 y)}^{2} + 30 x y - 30 x y = 181$
$\Rightarrow {(3 x)}^{2} + {(5 y)}^{2} + 2 (5 x) (6 y) = 181 + 30 x y$
$\Rightarrow {(3 x + y)}^{2} = 181 + 30 (- 6) = 181 - 180$
$\Rightarrow {(3 x + y)}^{2} = 1$
$\Rightarrow 3 x + y = \pm$ 1

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