Example 9 Under the conditions $x+2 y \leqslant 3, x \geqslant 0, y \geqslant 0$, the maximum value that $2 x+y$ can reach is $\qquad$
(2000, Hope Cup Junior High School Mathematics Competition Second Trial)
Solution: As shown in Figure 1, draw the line $x + 2y = 3$. The set of points satisfying the inequalities $x \geqslant 0, y \geqslant 0, x + 2y \leqslant 3$ is the region $\triangle ABO$ (including the boundaries) enclosed by the line and the $x$ and $y$ axes. To find the maximum value of $s = 2x + y$, we transform $s = 2x + y$ into $y = -2x + s$, whose corresponding image is a bundle of parallel lines with a slope of -2. To find the maximum value of $s$, it is converted into finding the maximum intercept of the parallel lines when they pass through $\triangle ABO$. Clearly, when the line $y = -2x + s$ passes through $A(3,0)$, the intercept $s$ is maximized, at which point $s = 6$.