给定两点A(0,0)与B(
)可以确定一条直线,该直线的斜率为:
![{\displaystyle m={\frac {y_{o}}{x_{o}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c7e9f2a14e6f5ed931d414812816cffecc9af79)
因此
![{\displaystyle y_{o}=mx_{o}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b779dafbafe96ee4edc32c37dc02d0744f90900)
对于在此条直线上的点C(
),可得斜率m
![{\displaystyle m={\frac {y-y_{o}}{x-x_{o}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea0d45f74769581d6d598ac770a59ce8d6bc3cc3)
![{\displaystyle y=y_{o}+m(x-x_{o})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2231317b4211f6494af1268fadf47eca215c934c)
通用的形式为
![{\displaystyle Ax=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d65b554c665205ea91902f56fc1836086c1fbeaf) |
|
![{\displaystyle Ax=C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6c8a6168d9556103fba14bff7d481b6643220ce) |
|
![{\displaystyle Ax+B=C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/929fcfbc732b9e2ddbf42529e48cc6178a5e6eb6) |
|
![{\displaystyle Ax+By=C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be59d3ad7986fd9f6ad8b476504a4f38926e095f) |
(当 时)
(当 时)
|
![{\displaystyle 2x+y=11\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/745b003101f250858a8d2984556477281a125c30)
![{\displaystyle -4x+3y=13\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6d2ac2e1ef8885a949b963746a4011e16ece907)
![{\displaystyle 2x+y=11\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/745b003101f250858a8d2984556477281a125c30)
![{\displaystyle -4x+3y=13\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6d2ac2e1ef8885a949b963746a4011e16ece907)
将第一行乘2,并加到第二行
![{\displaystyle 4x+2y=22\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98696540b736105d7418b5b59cf9611690f2c8d0)
![{\displaystyle -4x+3y=13\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6d2ac2e1ef8885a949b963746a4011e16ece907)
可得
=> ![{\displaystyle y=7}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48e26058942598c77301b34cf323650fe73a13fe)
将第一行乘3,并加到第二行
![{\displaystyle -6x-3y=-33\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e363f4f3b90386448f495a88cd6c90079b6ea8f)
![{\displaystyle -4x+3y=13\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6d2ac2e1ef8885a949b963746a4011e16ece907)
可得
=> ![{\displaystyle x=2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f39b6e42e5ffb81ac7b051b9e48b9a91d0713c7)
如果你看到一个像这样的线性方程组
![{\displaystyle 2x+y=11\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/745b003101f250858a8d2984556477281a125c30)
![{\displaystyle -4x+3y=13\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6d2ac2e1ef8885a949b963746a4011e16ece907)
你可以将第一行通过移项得到
![{\displaystyle y=-2x+11\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa0cb1629fdc43ea7c7331e9191cf2b5c375bc2)
之后,你可以替换这一项到第二行,从而可得
![{\displaystyle -4x+3(-2x+11)=13\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa8ee52eb03862b02948146c6f9ee0eeb5299949)
![{\displaystyle -4x-6x+33=13\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55c4a6d577d0da866af1d3a9465c00e7a0116ea2)
![{\displaystyle -10x+33=13\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a83f315ee32a12fbd3a343f375e504e22b0720e4)
![{\displaystyle -10x=-20\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6ef515da29cfd04e86ed7fc8e1fb9b09283f8cc)
![{\displaystyle x=2\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/850fca0fe30afa6fc6e912eec7b0d70616d98880)
之后,你可以将 x = 2 代替到原方程组中的任一个方程,从而解得 y = 7。通常,当原方程组中有一项单独的y时,使用这个方法更加简单。
如果你看到了一个像这样的方程组
![{\displaystyle 2x+y=11\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/745b003101f250858a8d2984556477281a125c30)
![{\displaystyle -4x+3y=13\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6d2ac2e1ef8885a949b963746a4011e16ece907)
用y来解
![{\displaystyle x+{\frac {1}{2}}y={\frac {11}{2}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a634fec34da89a0f8b8638d7fef3339a5efa162)
![{\displaystyle x+{\frac {3}{-4}}y={\frac {13}{-4}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f9e5be0e558773e731915b3a178990b93739f18)
![{\displaystyle y({\frac {1}{2}}-{\frac {3}{4}})={\frac {11}{2}}-{\frac {13}{-4}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07170e6f9157c1be2c9bb6313bb926f94388f2e0)
![{\displaystyle y={\frac {{\frac {11}{2}}-{\frac {13}{-4}}}{{\frac {1}{2}}-{\frac {3}{4}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c495f2e20ed6b5f6d24eb08a39ba97d962e60d1a)
用x来解
![{\displaystyle 2x+y=11\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/745b003101f250858a8d2984556477281a125c30)
![{\displaystyle {\frac {-4}{3}}x+y={\frac {13}{3}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12ea25ec9910fc5831d1f548704a33185baff0bc)
![{\displaystyle x(2-{\frac {-4}{3}})=11-{\frac {13}{3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c004c344bdf4de2f4869399c98b523ac5e1c4d86)
![{\displaystyle x={\frac {{\frac {11}{2}}-{\frac {13}{-4}}}{{\frac {1}{2}}-{\frac {3}{4}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd17a522b9754ad7082c79ee0dbae04af7f96e35)
![{\displaystyle A_{1}x+B_{1}y=C_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea2e19e17ab46d553b327234bfc5aedaa0b39f8f)
![{\displaystyle A_{2}x+B_{2}y=C_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/257389206014f2667b66ffe4e6348ec4739d4cbd)
消去变量x
![{\displaystyle x+{\frac {B_{1}}{A_{1}}}y={\frac {C_{1}}{A_{1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1edb8539d7986ef410440dd862fc4fbfd99c185b)
![{\displaystyle x+{\frac {B_{2}}{A_{2}}}y={\frac {C_{2}}{A_{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b62f0c59ee7ce47a235c95a160a7ad4342388c4d)
二式相减
![{\displaystyle y({\frac {B_{1}}{A_{1}}}-{\frac {B_{2}}{A_{2}}})={\frac {C_{1}}{A_{1}}}-{\frac {C_{2}}{A_{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f19406117e55672f932033714e7483e469b9bdf7)
用y来解
![{\displaystyle y={\frac {{\frac {C_{1}}{A_{1}}}-{\frac {C_{2}}{A_{2}}}}{{\frac {B_{1}}{A_{1}}}-{\frac {B_{2}}{A_{2}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/daff305e4ab618f31f8e51645116316073d81861)
消去变量y
![{\displaystyle {\frac {A_{1}}{B_{1}}}x+y={\frac {C_{1}}{B_{1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29b2bb872b26e185d0ed7757e4c4138c08e33b84)
![{\displaystyle {\frac {A_{2}}{B_{2}}}x+y={\frac {C_{2}}{B_{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba61952bb22572864e703d48286cac8f9add54ae)
二式相减
![{\displaystyle x({\frac {A_{1}}{B_{1}}}-{\frac {A_{2}}{B_{2}}})={\frac {C_{1}}{B_{1}}}-{\frac {C_{2}}{B_{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8335c57579887709fad3290e3f5d6b4bb3e1fb55)
用x来解
![{\displaystyle x={\frac {{\frac {C_{1}}{B_{1}}}-{\frac {C_{2}}{B_{2}}}}{{\frac {A_{1}}{B_{1}}}-{\frac {A_{2}}{B_{2}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37f05deaeca62dc71113e1278ddb8cb108992c1a)