<s>[INST] System of $3$ nonlinear equations 
Find positive solutions to the following:

\begin{align*}
x^2+y^2+xy=&1\\
y^2+z^2+yz=&3\\
z^2+x^2+xz=&4
\end{align*}
I simplified and got $x+y+z=\sqrt{7}$ and $x^2+y^2+xy=1$. How do I continue?
 [/INST] All positive real solutions are given by
$$
(x,y,z)=\frac{1}{\sqrt{7}}(2,1,4).
$$
I obtained the solutions by finding one linear equation (note that $x+y+z= 7$ does not hold), namely $2y + z - 3x=0$.
Then replacing $z=3x-2y$ yields two quadratic equations, which are easy to solve with the formula for quadratic equations. This gives all solutions, namely 
$$
(x,y,z)=\pm\frac{1}{\sqrt{7}}(2,1,4), \pm (0,-1,2).
$$
</s>