Find the value of $27 x^{3} + 8 y^{3}$ , if

(i) 3x + 2y = 14 and xy = 8

(ii) 3x + 2y = 20 and xy = $\frac{14}{9}$

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Solution

(i) 3x + 2y = 14 and xy = 8

Cubing both sides,

$(3 x + 2 y)^{3} = (14)^{3} \Rightarrow (3 x)^{3} + (2 y)^{3} + 3 \times 3 x \times 2 y (3 x + 2 y) = 2744 \Rightarrow 27 x^{3} + 8 y^{3} + 18 x y (3 x + 2 y) = 2744 \Rightarrow 27 x^{3} + 9 y^{3} + 18 \times 8 \times 14 = 2744 \Rightarrow 27 x^{3} + 8 y^{3} + 2016 = 2744 ∴ 27 x^{3} + 8 y^{3} = 2744 - 2016 = 728 (i i) 3 x + 2 y = 20 a n d x y = \frac{14}{9} Cubing both sides (3 x + 2 y)^{3} = (20)^{3} \Rightarrow (3 x)^{3} + (2 y)^{3} + 3 \times 3 x \times 2 y (3 x + 2 y) = 8000 \Rightarrow 27 x^{3} + 8 y^{3} + 8 \times 2 x y (3 x + 2 y) = 8000 \Rightarrow 27 x^{3} + 8 y^{3} + 18 \times \frac{14}{9} \times 20 = 8000 \Rightarrow 27 x^{3} + 8 y^{3} + 560 = 8000 \Rightarrow 27 x^{3} + 8 y^{3} = 8000 - 560 = 7440 ∴ 27 x^{3} + 8 y^{3} = 7440$

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