HMMT 2011 U17

Given positive real numbers $x, y$, and $z$ that satisfy the following system of equations:

$$\begin{aligned} x^{2}+y^{2}+x y & =1, \\ y^{2}+z^{2}+y z & =4, \\ z^{2}+x^{2}+z x & =5, \end{aligned}$$

find $x+y+z$.