Example 6. Let the point $(x, y)$ vary within the set $\left\{\begin{array}{l}y-x+1 \geqslant 0, \\ 2 x-y-3 \geqslant 0\end{array}\right.$, find the minimum value of $x+2 y$.
This problem originally belongs to the category of quantitative relationship, but the given set $\left\{\begin{array}{l}y-x+1 \geqslant 0, \\ 2 x-y-3 \geqslant 0\end{array}\right.$ can be represented on a coordinate plane, as shown in Figure 4. If we let $x+2 y=k$, then $y=-\frac{1}{2} x+\frac{k}{2}$, which represents a family of parallel lines with a slope of $-\frac{1}{2}$. Thus, the original problem is transformed into finding when the y-intercept is minimized as 1 varies over the region $G$. Since the range of region $G$ is $x-1 \leqslant y \leqslant 2 x-3, 2 \leqslant x<+\infty$, it is clear that the y-intercept of the family of parallel lines? is minimized at the point $(2,1)$. Therefore, the minimum value of $\mathrm{x}+2 \mathrm{y}$ is
$$
\mathrm{k}=2+2 \times 1=4
$$