#### 線性方程祖求解方法

$\left\{\begin{array}{rrrrrll}x& +& y& -& z& =1& \phantom{\rule{30px}{0ex}}{\text{(1)}}\\ 8x& +& 3y& -& 6z& =1& \phantom{\rule{30px}{0ex}}{\text{(2)}}\\ -4x& -& y+& 3z& =1& \phantom{\rule{30px}{0ex}}{\text{(3)}}\end{array}$
##### 消元法

$\begin{array}{rrrrrll}x& +& y& -& z& =1& \phantom{\rule{30px}{0ex}}{\text{(1)}}\\ -4x& -& y& +& 3z& =1& \phantom{\rule{30px}{0ex}}{\text{(3)}}\end{array}\phantom{\rule{0ex}{0ex}}\text{------------------------}\phantom{\rule{30px}{0ex}}+\phantom{\rule{0ex}{0ex}}\begin{array}{rrrrrll}-3x& \phantom{+}& \phantom{0}& +& 2z& =2& \phantom{\rule{30px}{0ex}}{\text{(4)}}\end{array}$

$\begin{array}{rrrrrlll}x& +& y& -& z& =1& \phantom{\rule{30px}{0ex}}{\text{(1)}}& \phantom{\rule{20px}{0ex}}{\text{× 3}}\\ 8x& +& 3y& -& 6z& =1& \phantom{\rule{30px}{0ex}}{\text{(2)}}& \end{array}$
$\begin{array}{rrrrrlll}3x& +& 3y& -& 3z& =3& \phantom{\rule{30px}{0ex}}{\text{(1)}}& \phantom{\rule{40px}{0ex}}\\ 8x& +& 3y& -& 6z& =1& \phantom{\rule{30px}{0ex}}{\text{(2)}}& \end{array}\phantom{\rule{0ex}{0ex}}\text{------------------------}\phantom{\rule{30px}{0ex}}-\phantom{\rule{50px}{0ex}}\phantom{\rule{0ex}{0ex}}\begin{array}{rrrrrlll}-5x& \phantom{+}& \phantom{0y}& +& 3z& =2& \phantom{\rule{30px}{0ex}}{\text{(5)}}& \phantom{\rule{60px}{0ex}}\end{array}$

$\begin{array}{rrrrrl}-3x& +& 2z& =2& \phantom{\rule{30px}{0ex}}{\text{(4)}}& \phantom{\rule{20px}{0ex}}{\text{× 3}}\\ -5x& +& 3z& =2& \phantom{\rule{30px}{0ex}}{\text{(5)}}& \phantom{\rule{20px}{0ex}}{\text{× 2}}\end{array}$
$\begin{array}{rrrrrl}-9x& +& 6z& =6& \phantom{\rule{30px}{0ex}}{\text{(4)}}& \phantom{\rule{40px}{0ex}}\\ -10x& +& 6z& =4& \phantom{\rule{30px}{0ex}}{\text{(5)}}& \end{array}\phantom{\rule{0ex}{0ex}}\text{------------------------}\phantom{\rule{30px}{0ex}}-\phantom{\rule{50px}{0ex}}\phantom{\rule{0ex}{0ex}}\begin{array}{rrrrrl}\phantom{+01}x& \phantom{+}& \phantom{0z}& =2& \phantom{\rule{30px}{0ex}}{\text{(6)}}& \phantom{\rule{45px}{0ex}}\end{array}$

$\begin{array}{rrrll}-3\left(2\right)& +& 2z& =2& \phantom{\rule{30px}{0ex}}{\text{(4)}}\\ -6& +& 2z& =2& \\ & & 2z& =2+6& \\ & & 2z& =8& \\ & & z& =8÷2& \\ & & z& =4& \end{array}$

$\begin{array}{rll}2+y-4& =1& \phantom{\rule{30px}{0ex}}{\text{(1)}}\\ y& =1-2+4& \\ y& =3& \end{array}$

##### 代入法

$\begin{array}{rl}x=1-y+z& \phantom{\rule{30px}{0ex}}{\text{(1)}}\end{array}$

$\begin{array}{rll}8\left(1-y+z\right)+3y-6z& =1& \phantom{\rule{30px}{0ex}}{\text{(2)}}\\ 8-8y+8z+3y-6z& =1& \\ -5y+2z& =1-8& \\ -5y+2z& =-7& \phantom{\rule{30px}{0ex}}{\text{(4)}}\end{array}$

$\begin{array}{rll}-4\left(1-y+z\right)-y+3z& =1& \phantom{\rule{30px}{0ex}}{\text{(3)}}\\ -4+4y-4z-y+3z& =1& \\ 3y-z& =1+4& \\ 3y-z& =5& \phantom{\rule{30px}{0ex}}{\text{(5)}}\end{array}$

$\begin{array}{rl}z=3y-5& \phantom{\rule{30px}{0ex}}{\text{(5)}}\end{array}$

$\begin{array}{rll}-5y+2\left(3y-5\right)& =-7& \phantom{\rule{30px}{0ex}}{\text{(4)}}\\ -5y+6y-\mathrm{10}& =-7& \\ y& =-7+10& \\ y& =3& \end{array}$

$\begin{array}{rll}z& =3\left(3\right)-5& \phantom{\rule{30px}{0ex}}{\text{(5)}}\\ z& =9-5& \\ z& =4& \end{array}$

$\begin{array}{rll}x& =1-3+4& \phantom{\rule{30px}{0ex}}{\text{(1)}}\\ x& =2& \end{array}$

##### 圖象法

$\left\{\begin{array}{rrrr}x& +& y& =3\\ 2x& -& y& =-3\end{array}$

##### 逆矩陣法

$\begin{array}{rl}AB& =C\\ \left(\begin{array}{rrr}1& 2& -1\\ 8& 3& -6\\ -4& -1& 3\end{array}\right)\left(\begin{array}{c}x\\ y\\ z\end{array}\right)& =\left(\begin{array}{c}1\\ 1\\ 1\end{array}\right)\end{array}$

$\begin{array}{rl}{A}^{-1}AB& ={A}^{-1}C\\ B& ={A}^{-1}C\end{array}$

$\begin{array}{rl}{A}^{-1}& =\left(\begin{array}{rrr}-3& 2& 3\\ 0& 1& 2\\ -4& 3& 5\end{array}\right)\\ B& =\left(\begin{array}{rrr}-3& 2& 3\\ 0& 1& 2\\ -4& 3& 5\end{array}\right)\left(\begin{array}{c}1\\ 1\\ 1\end{array}\right)\\ B& =\left(\begin{array}{c}2\\ 3\\ 4\end{array}\right)\end{array}$

##### 高斯消元法／高斯－若爾當消元法

$A=\left(\begin{array}{rrrrr}1& 1& -1& \text{|}& 1\\ 8& 3& -6& \text{|}& 1\\ -4& -1& 3& \text{|}& 1\end{array}\right)$

$A=\left(\begin{array}{rccll}1& 0.375& -0.75& \text{|}& 0.125\\ 0& 1& -0.4& \text{|}& 1.4\\ 0& 0& 1& \text{|}& 4\end{array}\right)$

$A=\left(\begin{array}{rrrrr}1& 0& 0& \text{|}& 2\\ 0& 1& 0& \text{|}& 3\\ 0& 0& 1& \text{|}& 4\end{array}\right)$