Example 4 In the border desert area, patrol vehicles travel 200 kilometers per day, and each vehicle can carry enough gasoline to travel for 14 days. There are 5 patrol vehicles that set out from base $A$ simultaneously, complete their tasks, and then return along the same route to the base. To allow 3 of them to patrol as far as possible and then return together, vehicles Jia and Yi travel to point $B$ along the way, leave behind the gasoline necessary for their own return to the base, and give the remaining gasoline to the other 3 vehicles to use. How far can these 3 vehicles travel in kilometers?
(1995, Hebei Province Junior High School Mathematics Competition)
Analysis: The key is to list the relationship formula for the distance traveled by the car, and then solve it by analyzing the characteristics of the formula.

Let 5 cars travel to point $B$ in $x$ days, and 3 cars continue to travel for $y$ days, with the distance $s=200(x+y)$, which is to find the maximum value of $s$. According to the problem, we have
$$
2[3(x+y)+2 x]=14 \times 5,
$$

which simplifies to $5 x+3 y=35$, and $x>0, y>0$.
Thus, $14 \times 5-(5+2) x \leqslant 14 \times 3, x \geqslant 4$, then $y \leqslant 5$.

Therefore, the maximum value of $s$ is obtained when $x=4, y=5$, which is $s=200 \times(4+5)=1800$ kilometers.