I2.4 Given that $x_{0}$ and $y_{0}$ are real numbers satisfying the system of equations $\left\{\begin{array}{l}y=\frac{T}{x} \\ y=|x|+T\end{array}\right.$.
If $W=x_{0}+y_{0}$, find the value of $W$.
$$
\begin{array}{l}
\left\{\begin{array}{l}
y=\frac{1}{x} \\
y=|x|+1
\end{array}\right. \\
\frac{1}{x}=|x|+1 \ldots \ldots \ldots .(*) \\
\frac{1}{x}=x+1 \text { or } \frac{1}{x}=-x+1 \\
1=x^{2}+x \text { or } 1=-x^{2}+x \\
x^{2}+x-1=0 \text { or } x^{2}-x+1=0 \\
x=\frac{-1 \pm \sqrt{5}}{2} \text { or no solution }
\end{array}
$$

Check: sub. $x=\frac{-1-\sqrt{5}}{2}$ into $(*)$
$$
\text { LHS }=\frac{2}{-1-\sqrt{5}}0 \text { (rejected) }
$$

When $x=\frac{-1+\sqrt{5}}{2} ;$ LHS $=\frac{2}{-1+\sqrt{5}}=\frac{\sqrt{5}+1}{2} ;$ RHS $=\frac{-1+\sqrt{5}}{2}+1=\frac{1+\sqrt{5}}{2} \quad$ (accepted)
$$
y=\frac{1+\sqrt{5}}{2} \Rightarrow W=x_{0}+y_{0}=\frac{-1+\sqrt{5}}{2}+\frac{1+\sqrt{5}}{2}=\sqrt{5}
$$