<s>[INST] How to solve this system of non-linear equations My knowledge of algebra is still insufficient to solve this problem. Any help in solving the system of equations would be greatly appreciated.
$$
xy(x+y)=30\\
x^3+y^3=35
$$
 [/INST] Note that
$$
(x+y)^3
 = x^3 + 3x^2y + 3xy^2 + y^3 = x^3+y^3 + 3xy(x+y)
 = 35 + 3 \cdot 30 = 125
$$
and therefore $x+y = \sqrt[3]{125} = 5$ and $xy = \frac{30}{x+y} = 6$.
Thus $y = 5-x$ and $6 = xy = x(5-x)$, and so $x^2-5x+6=0$, which implies $x \in \{2,3\}$ and so $y \in \{3,2\}$. Thus, either $(x,y)=(2,3)$ or $(x,y) = (3,2)$...
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