If $2 x + 3 y = 13$ and $x y = 6$ , find the value of $8 x^{3} + 21 y^{3}$ .

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Solution

$2 x + 3 y = 13, x y = 6$
Cubing both sides,
$(2 x + 3 y)^{3} = (13)^{2}$
$\Rightarrow (2 x)^{3} + (3 y)^{3} + 3 \times 2 x \times 3 y (2 x + 3 y) = 2197$
$\Rightarrow 8 x^{3} + 21 y^{3} + 18 x y (2 x + 3 y) = 2197$
$\Rightarrow 8 x^{3} + 21 y^{3} + 18 \times 6 \times 13 = 2197$
$\Rightarrow 8 x^{3} + 21 y^{3} + 1404 = 2197$
$\Rightarrow 8 x^{3} + 21 y^{3} = 2197 - 1404 = 793$
$∴ 8 x^{3} + 21 y^{3} = 793$

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