Example 6: 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 3.15 yuan; if you buy 4 pieces of A, 10 pieces of B, and 1 piece of C, it costs 4.20 yuan. Now, if you buy 1 piece each of A, B, and C, it will cost $\qquad$ yuan.
(1985. National Junior High School Mathematics League)
Solution: Let the cost of 1 item of A be $x$ yuan, 1 item of B be $y$ yuan, and 1 item of C be $z$ yuan, then the cost of 1 item each of A, B, and C is $(x+y+z)$ yuan. From the given conditions, we have
$$
\begin{array}{l}
3 x+7 y+z=3.15, \\
4 x+10 y+z=4.20 .
\end{array}
$$

Let
$$
\begin{array}{l}
x+y+z \\
=a(3 x+7 y+z)+b(4 x+10 y+z) \\
=(3 a+4 b) x+(7 a+10 b) y+(a+b) z .
\end{array}
$$

By comparing the coefficients of like terms on both sides of the equation, we get
$$
\left\{\begin{array}{l}
3 a+4 b=1, \\
7 a+10 b=1, \\
a+b=1 .
\end{array}\right.
$$

Solving (1) and (3), we get $a=3, b=-2$. Substituting into (2), it fits.
$$
\therefore x+y+z=3 \times 3.15-2 \times 4.20=1.05 \text {. }
$$