1. Given real numbers $x_{0}, y_{0}$ are solutions to the system of equations $\left\{\begin{array}{l}y=\frac{1}{x}, \\ y=|x|+1\end{array}\right.$. Then $x_{0}+y_{0}=$ $ـ$. $\qquad$
$$
=\ldots 1 . \sqrt{5} .
$$

From the given, $\frac{1}{x}=|x|+1$, since $x \neq 0$, we get (1) $\left\{\begin{array}{l}x<0, \\ x^{2}+x+1=0 .\end{array}\right.$ Equation (I) has no solution.
Solving (I) yields $x_{0}=\frac{\sqrt{5}-1}{2}$, thus we get $y_{0}=\frac{\sqrt{5}+1}{2}$.
$$
\therefore x_{0}+y_{0}=\sqrt{5} \text {. }
$$