Solve the following systems of equations.
$x + y = 9, x^{1 / 3} + y^{1 / 3} = 3$

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Solution

$x + y = 9$

$⟹ (x^{1 / 3})^{3} + (y^{1 / 3})^{3} = 9$

$(x^{1 / 3} + y^{1 / 3}) (x^{2 / 3} + y^{2 / 3} - x^{1 / 3} y^{1 / 3}) = 9$

$⟹ (x^{1 / 3} + y^{1 / 3}) {(x^{1 / 3} + y^{1 / 3})^{2} - 3 x^{1 / 3} y^{1 / 3})} = 9$

$⟹ 3 (3^{2} - 3 x^{1 / 3} y^{1 / 3}) = 9$

$⟹ x^{1 / 3} y^{1 / 3} = 2$

$⟹ x y = 8$

$(x - y)^{2} = (x + y)^{2} - 4 x y = 9^{2} - 4.8 = 49$

$⟹ x - y = 7$

And, $x + y = 9$

On solving-

$x = 8, y = 1$

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