Ralph went to the store and bought 12 pairs of socks for a total of $24. Some of the socks he bought cost$1 a pair, some of the socks he bought cost $3 a pair, and some of the socks he bought cost$4 a pair. If he bought at least one pair of each type, how many pairs of $1 socks did Ralph buy?

A. 4

B. 5

C. 6

D. 7

E. 8

1. 2015 AMC 8 官方解析

设一元一双、三元一双和四元一双的袜子分别是 $x$ 双、$y$ 双、$z$ 双，$\begin{cases} x + y + z = 12 \\ x + 3y + 4z = 24 \end{cases}$ 得出：$2y + 3z = 12$，$\begin{cases} y = 0 \\ z = 4 \end{cases}$、$\begin{cases} y = 3 \\ z = 2 \end{cases}$、$\begin{cases} y = 6 \\ z = 0 \end{cases}$，其中只有 $\begin{cases} y = 3 \\ z = 2 \end{cases}$ 满足条件，所以 $x = 12 - 3 - 2 = 7$。