解：由$6x + y = 1$得$y = 1 - 6x，$
将$y = 1 - 6x$代入$12x - 3y = 7，$得$12x - 3(1 - 6x)=7，$
$12x - 3 + 18x = 7，$
$30x = 10，$
解得$x=\frac{1}{3}。$
把$x=\frac{1}{3}$代入$y = 1 - 6x，$得$y = 1 - 6\times\frac{1}{3}=1 - 2=-1。$
所以方程组的解为$\begin{cases}x=\frac{1}{3}\\y = - 1\end{cases}。$
