<s>[INST] How can I solve this system? Having this system:
$$\begin{cases}
2 x(x^2 + y - 11) + x + y^2 - 7 = 0,\\
x^2 + y - 11 + 2 y (x + y^2 - 7) = 0 .
\end{cases}$$
How can I solve it without using a computer?. 
Thank you!
 [/INST] Let $a = x^2+y-11$ and $b = x+y^2-7$.
The equations we get are as follows:
$$2ax+b=0$$
$$a+2by=0$$
Replacing $a$ in the first equation yields:
$$-4bxy+b = 0$$
which gives us two options: $1-4xy = 0$ ($x = 1/4y$) or $b = 0$ ($x = 7-y^2)$. In both cases, we have expressed $x$ explicitly, we can insert it back the second original equation and solve it.
1.) $(1/4y)^2 + y - 11 + 2y(1/4y + y^2 -7) = 0$
that is (hopefully) equivalent to
$$ 32y^5 -14y^4+16y^3-21\cdot 18y^2+1 = 0$$ 
which looks painful.
2.) $(7-y)^2 + y - 11 = 0$ which is easily solvable.
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