If $3 x - 2 y = 13$ and $x y = 5$ , find the value of $27 x^{3} - 8 y^{3}$

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Solution

$3 x - 2 y = 13$
$⟹ {(3 x - 2 y)}^{2} = 13^{2}$
$⟹ 9 x^{2} + 4 y^{2} - 12 x y = 169$
$⟹ 9 x^{2} + 4 y^{2} + 6 x y = 169 + 18 x y = 169 + 90 = 359$
$27 x^{3} - 8 y^{3} = {(3 x)}^{3} - {(2 y)}^{3}$
$= (3 x - 2 y) (9 x^{2} + 4 y^{2} + 6 x y)$
$= (13) (359) = 4667$

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