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方程组.
方程组(英語:system of equations)又称联立方程(simultaneous equations),是两个或两个以上含有多个未知数的方程联立得到的集。未知数的值称为方程组的根,求方程组根的过程称为解方程组。一般在方程式的左边加大括号标注。
解方程组的方法
解方程组的方法大致上有画图法、代入法、消元法(包括高斯消元法)、矩阵法(包括克莱姆法则)等。
画图法
画图法就是把两条方程式画在图上,两线的交点就是解了。
如要解决以下方程组︰
![{\displaystyle {\begin{cases}2x+y=8\\x+y=6\end{cases))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dcb71297744ea0ec9d4f2b1a0dd2fac218fc2c59)
首先要把要把它们画在图上︰
|
绿色为 ,
红色为 。
|
两线的交点是︰
![{\displaystyle (x,y)=(2,4)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3ed9c97a74c00bbd635374bc1b5c00d7e42c1e5)
所以它的解为:
![{\displaystyle {\begin{cases}x=2\\y=4\end{cases))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d0a2d129c696e8d72f3ce5132470f281bcf7b0a)
代入消元法
如要解决以下方程组︰
![{\displaystyle {\begin{cases}2x+y=8\\x+y=6\end{cases))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dcb71297744ea0ec9d4f2b1a0dd2fac218fc2c59)
过程是︰
|
|
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然后把
代入到其中一条方程式里︰
![{\displaystyle {\begin{aligned}y&=6-x\\&=6-(2)\\&=4\end{aligned))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae937e7daeed5680f7779cac505ab468fe8c9abe)
所以它的解为:
![{\displaystyle {\begin{cases}x=2\\y=4\end{cases))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d0a2d129c696e8d72f3ce5132470f281bcf7b0a)
加减消元法
如要解决以下方程组︰
![{\displaystyle {\begin{cases}2x+y=8\\x+y=6\end{cases))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dcb71297744ea0ec9d4f2b1a0dd2fac218fc2c59)
把两个相减︰
![{\displaystyle {\begin{aligned}\ 2x+y=8\\{\underline {-)\ x+y=6))\\({\text{subtract)))\ x=2\end{aligned))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e51fb157f5ed4775decca6328003bb834fbdb0c)
然后把
代入到其中一条方程式里︰
![{\displaystyle {\begin{aligned}x+y&=6&({\text{the second equation)))\\(2)+y&=6\\y&=6-2\\y&=4\end{aligned))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d0e5351080ebf42c5bb6c58d16490af2118f765)
所以它的解为:
![{\displaystyle {\begin{cases}x=2\\y=4\end{cases))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d0a2d129c696e8d72f3ce5132470f281bcf7b0a)
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