One, (20 points) A woodworking factory produces two types of chairs, A and B. Each chair must go through two processes, woodworking and painting, to be completed. The daily working hours of the woodworkers and painters and the time required to make (paint) one chair are shown in Table 1. It is known that the factory earns a profit of 2 yuan for selling one chair of type A and 3 yuan for selling one chair of type B. If one woodworker and one painter form a group, how many chairs of types A and B should each group make daily to maximize profit?
Table 1
\begin{tabular}{|c|c|c|c|}
\hline \multirow{2}{*}{ Worker Type } & \multicolumn{2}{|c|}{ Time Required per Chair } & \multirow{2}{*}{ Daily Working Hours } \\
\cline { 1 - 3 } & A & B & \\
\hline Woodworker & 1 & 2 & 8 \\
\hline Painter & 3 & 1 & 9 \\
\hline
\end{tabular}
Let's do $x$ chairs of type A and $y$ chairs of type B each day, with daily profit being $M$, then we have
$$
\left\{\begin{array}{l}
x+2 y \leqslant 8, \\
3 x+y \leqslant 9, \\
M=2 x+3 y, \\
x \geqslant 0, y \geqslant 0 .
\end{array}\right.
$$

From Figure 7, it is clear that,
$$
\text { when } x=2, y=3
$$

$M$ is maximized, with its maximum value being 13.