$$ \begin{cases} a_{11}x_{1}+a_{12}x_{2}+\cdots+a_{1n}x_{n}=b_{1}\\ a_{21}x_{1}+a_{22}x_{2}+\cdots+a_{2n}x_{n}=b_{2}\\ \vdots\\ a_{n1}x_{1}+a_{n2}x_{2}+\cdots+a_{nn}x_{n}=b_{n} \end{cases} $$

$$ \begin{array}{ll} &\begin{cases} x+y=3&\cdots(1)\\ x-y=1&\cdots(2)\\ \end{cases}\\ &\begin{cases} 1x+1y=3&\cdots(1)\\ 1x+(-1)y=1&\cdots(2)\\ \end{cases}\\ (2)+(-1)\times(1)\to(2)'&\\ &\begin{cases} 1x+1y=3&\cdots(1)\\ 0x+(-2)y=-2&\cdots(2)'\\ \end{cases}\\ (2)'\div(-2)\to(2)''&\\ &\begin{cases} 1x+1y=3&\cdots(1)\\ 0x+1y=1&\cdots(2)''\\ \end{cases}\\ (1)+(-1)\times(2)''\to(1)'&\\ \end{array} $$