定義： | a b c d | = a d − b c {\displaystyle {\begin{vmatrix}a&b\\c&d\end{vmatrix}}=ad-bc}
例如： | 2 3 1 4 | = 2 ⋅ 4 − 3 ⋅ 1 = 5 {\displaystyle {\begin{vmatrix}2&3\\1&4\end{vmatrix}}=2\cdot 4-3\cdot 1=5}
方程式組為： { a 1 x + b 1 y = c 1 a 2 x + b 2 y = c 2 {\displaystyle {\begin{cases}a_{1}x+b_{1}y=c_{1}\\a_{2}x+b_{2}y=c_{2}\end{cases}}}
其解如下： x = | c 1 b 1 c 2 b 2 | | a 1 b 1 a 2 b 2 | , y = | a 1 c 1 a 2 c 2 | | a 1 b 1 a 2 b 2 | {\displaystyle x={\frac {\left|{\begin{matrix}c_{1}&b_{1}\\c_{2}&b_{2}\end{matrix}}\right|}{\left|{\begin{matrix}a_{1}&b_{1}\\a_{2}&b_{2}\end{matrix}}\right|}},\qquad y={\frac {\left|{\begin{matrix}a_{1}&c_{1}\\a_{2}&c_{2}\end{matrix}}\right|}{\left|{\begin{matrix}a_{1}&b_{1}\\a_{2}&b_{2}\end{matrix}}\right|}}}
或是直接套公式：
x = c 1 b 2 − c 2 b 1 a 1 b 2 − a 2 b 1 {\displaystyle x={\frac {c_{1}b_{2}-c_{2}b_{1}}{a_{1}b_{2}-a_{2}b_{1}}}} ,
y = a 1 c 2 − a 2 c 1 a 1 b 2 − a 2 b 1 {\displaystyle y={\frac {a_{1}c_{2}-a_{2}c_{1}}{a_{1}b_{2}-a_{2}b_{1}}}}
例：
{ 2 x + 37 y = 2 2 x + 119 y = 2 {\displaystyle {\begin{cases}2x+37y=2\\2x+119y=2\end{cases}}}
解： x = 2 × 119 − 2 × 37 2 × 119 − 2 × 37 = 238 − 74 238 − 74 {\displaystyle x={\frac {2\times 119-2\times 37}{2\times 119-2\times 37}}={\frac {238-74}{238-74}}} ,
238－74=164, x = 1
y = 4 − 4 238 − 74 {\displaystyle y={\frac {4-4}{238-74}}}
4－4=0,238－74=164, y = 0
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