Example 7 A home appliance manufacturing company, based on market research analysis, has decided to adjust its production plan and is preparing to produce a total of 360 units of air conditioners, color TVs, and refrigerators per week (calculated at 120 labor hours). Moreover, the production of refrigerators should be at least 60 units. It is known that the labor hours required per unit and the output value per unit of these home appliances are as shown in Table 1.
Table 1
\begin{tabular}{|l|c|c|c|}
\hline Home Appliance Name & Air Conditioner & Color TV & Refrigerator \\
\hline Labor Hours & $\frac{1}{2}$ & $\frac{1}{3}$ & $\frac{1}{4}$ \\
\hline Output Value (thousand yuan) & 4 & 3 & 2 \\
\hline
\end{tabular}

How many units of air conditioners, color TVs, and refrigerators should be produced each week to maximize the output value? What is the maximum output value (in thousand yuan)?
(12th Jiangsu Province Junior High School Mathematics Competition)
Solution: Let the number of air conditioners produced each week be $x$, the number of color TVs be $y$, and the number of refrigerators be $z$, with the total output value being $w$ thousand yuan. According to the problem, we have
$$
\left\{\begin{array}{l}
x+y+z=360, \\
\frac{1}{2} x+\frac{1}{3} y+\frac{1}{4} z=120, \\
w=4 x+3 y+2 z, \\
z \geqslant 60 .
\end{array}\right.
$$

From (1) and (2), we get
$$
x=\frac{z}{2}, y=360-\frac{3 z}{2} \text {, }
$$

Substituting into (3), we get
$$
w=-\frac{1}{2} z+1080 .
$$

Since $k=-\frac{1}{2}<0$,
$w$ increases as $z$ decreases.
Given $z \geqslant 60$, when $z=60$, $w$ is maximized, with the maximum value being 1050.
At this point, $x=30, y=270$.
Therefore, to maximize the output value, 30 air conditioners, 270 color TVs, and 60 refrigerators should be produced each week, with the highest output value being 1050 thousand yuan.