Example 7 A factory produces two types of products, A and B. The production of each ton of product requires electricity (kWh), coal (tons), labor (people), and output value (thousand yuan) as shown in the table below:
\begin{tabular}{|c|c|c|c|c|}
\hline Type & Electricity & Coal & Labor & Output Value \\
\hline A & 2 & 3 & 5 & 7 \\
\hline B & 8 & 5 & 2 & 10 \\
\hline
\end{tabular}

It is known that the factory's full labor capacity is 200 people, and according to the limits, the daily electricity usage cannot exceed 160 thousand kWh, and the coal usage cannot

exceed 150 tons. How should the production quantities of these two products be arranged to achieve the maximum output value?
Solution: Let the production of products A and B be $x, y$ tons, respectively. According to the problem, we have
$$
\left\{\begin{array}{l}
2 x+8 y \leqslant 160, \\
3 x+5 y \leqslant 150, \\
5 x+2 y \leqslant 200, \\
x \geqslant 0, \\
y \geqslant 0 .
\end{array}\right.
$$

In the Cartesian coordinate system, draw the three lines:
$2 x+8 y=160, 3 x+5 y=150, 5 x+2 y=200$, as shown in Figure 2. The set of points that satisfy the constraints of the inequality system is the pentagonal region (including the boundaries) formed by the intersection of the three lines and the $x, y$ axes. The output value $s=7 x+10 y$ can be transformed into
$$
y=-\frac{7}{10} x+\frac{s}{10},
$$

whose corresponding graph is a system of parallel lines with a slope of $-\frac{7}{10}$. To find the maximum value of $s$, it is converted into finding the maximum intercept of the parallel line system when it passes through the pentagon.

Obviously, when the line in the parallel line system passes through the intersection point $\left(\frac{700}{19}, \frac{150}{19}\right)$ of the lines $3 x+5 y=150$ and $5 x+2 y=200$, i.e., when products A and B are produced $\frac{700}{19}$ tons and $\frac{150}{19}$ tons, respectively, the maximum value of $s$ is obtained, $s_{\text {max }}=336 \frac{16}{19}$ (thousand units).