Example 2 There are three types of goods, A, B, and C. If you buy 3 pieces of A, 7 pieces of B, and 1 piece of C, it costs a total of 315 yuan; if you buy 4 pieces of A, 10 pieces of B, and 1 piece of C, it costs a total of 420 yuan. Question: How much would it cost to buy one piece each of A, B, and C?
Let the unit prices of A, B, and C be $x$, $y$, and $z$ yuan, respectively. Then, according to the problem, we have
$$
\left\{\begin{array}{l}
3 x+7 y+z=315, \\
4 x+10 y+z=420 .
\end{array}\right.
$$

The problem actually only requires finding the value of $x+y+z$, without necessarily solving for $x$, $y$, and $z$ individually. Therefore, we should try to isolate the value of $x+y+z$.
The system of equations (1) can be equivalently transformed into
$$
\left\{\begin{array}{l}
2(x+3 y)+(x+y+z)=315, \\
3(x+3 y)+(x+y+z)=420 .
\end{array}\right.
$$

It is easy to see that, $x+y+z=105$.
Therefore, purchasing one each of A, B, and C would cost 105 yuan.