Question:
If $2 x+3 y=13$ and $x y=6$, Find the value of $8 x^{3}+27 y^{3}$
Solution:
Given, 2x + 3y = 13, xy = 6
We know that,
$(2 x+3 y)^{3}=13^{2}$
$\Rightarrow 8 x^{3}+27 y^{3}+3(2 x)(3 y)(2 x+3 y)=2197$
$\Rightarrow 8 x^{3}+27 y^{3}+18 x y(2 x+3 y)=2197$
Substitute 2x + 3y = 13, xy = 6
$\Rightarrow 8 x^{3}+27 y^{3}+18(6)(13)=2197$
$\Rightarrow 8 x^{3}+27 y^{3}+1404=2197$
$\Rightarrow 8 x^{3}+27 y^{3}=2197-1404$
$\Rightarrow 8 x^{3}+27 y^{3}=793$
Hence, the value of $8 x^{3}+27 y^{3}=793$
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