1. The system of equations in $x, y, z$
$$
\left\{\begin{array}{l}
x y+y z+z x=1, \\
5 x+8 y+9 z=12
\end{array}\right.
$$

all real solutions $(x, y, z)$ are
$$
\text { II. 1. }\left(1, \frac{1}{2}, \frac{1}{3}\right) \text {. }
$$

Substituting equation (2) into equation (1) yields
$$
\begin{array}{l}
5 x^{2}+8 y^{2}+4 x y-12 x-12 y+9=0 \\
\Rightarrow(4 y+x-3)^{2}+9(x-1)^{2}=0 \\
\Rightarrow x-1=0,4 y+x-3=0 \\
\Rightarrow(x, y, z)=\left(1, \frac{1}{2}, \frac{1}{3}\right) .
\end{array}
$$