Формулы деления отрезка в отношении
![\lambda:](https://tex.z-dn.net/?f=%5Clambda%3A)
![x=\frac{x_1+\lambda x_2}{1+\lambda } ,y=\frac{y_1+\lambda y_2}{1+\lambda }, z=\frac{z_1+\lambda z_2}{1+\lambda }\\\lambda =\frac{3}{5}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7Bx_1%2B%5Clambda+x_2%7D%7B1%2B%5Clambda+%7D+%2Cy%3D%5Cfrac%7By_1%2B%5Clambda+y_2%7D%7B1%2B%5Clambda+%7D%2C+z%3D%5Cfrac%7Bz_1%2B%5Clambda+z_2%7D%7B1%2B%5Clambda+%7D%5C%5C%5Clambda+%3D%5Cfrac%7B3%7D%7B5%7D)
Сначала делим в отношении 3/5 от точки А.
![y=\frac{5-\frac{3}{5}}{1+\frac{3}{5}}=5, z=\frac{5+\frac{3}{5}3}{1+{\frac{3}{5}}}=\frac{34}{8}=\frac{17}{4}](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B5-%5Cfrac%7B3%7D%7B5%7D%7D%7B1%2B%5Cfrac%7B3%7D%7B5%7D%7D%3D5%2C+z%3D%5Cfrac%7B5%2B%5Cfrac%7B3%7D%7B5%7D3%7D%7B1%2B%7B%5Cfrac%7B3%7D%7B5%7D%7D%7D%3D%5Cfrac%7B34%7D%7B8%7D%3D%5Cfrac%7B17%7D%7B4%7D)
Точка В(-1,5,17/4)
Теперь делим ,начиная от точки Р.
![x=\frac{-1-\frac{3}{5}}{1+\frac{3}{5}}=-1, y=\frac{-1+3}{1+\frac{3}{5}}=\frac{5}{4}, z=\frac{3+3}{1+\frac{3}{5}}=\frac{15}{4}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-1-%5Cfrac%7B3%7D%7B5%7D%7D%7B1%2B%5Cfrac%7B3%7D%7B5%7D%7D%3D-1%2C+y%3D%5Cfrac%7B-1%2B3%7D%7B1%2B%5Cfrac%7B3%7D%7B5%7D%7D%3D%5Cfrac%7B5%7D%7B4%7D%2C+z%3D%5Cfrac%7B3%2B3%7D%7B1%2B%5Cfrac%7B3%7D%7B5%7D%7D%3D%5Cfrac%7B15%7D%7B4%7D)
Точка С(-1,5/4,15/4)
Составим матрицу и приведем её в ступенчатый вид:
![\displaystyle \left[\begin{array}{cccc}1&2&1&3\\1&2&-2&-3\\2&-3&-1&0\end{array}\right] \rightarrow R_2-R_1\rightarrow \left[\begin{array}{cccc}1&2&1&3\\0&0&-3&-6\\2&-3&-1&0\end{array}\right]\rightarrow \\\\\rightarrow R_3-2R_1 \rightarrow \left[\begin{array}{cccc}1&2&1&3\\0&0&-3&-6\\0&-7&-3&-6\end{array}\right] \rightarrow R_3\leftrightarrow R_2\rightarrow \\\\\left[\begin{array}{cccc}1&2&1&3\\0&-7&-3&-6\\0&0&-3&-6\end{array}\right]\rightarrow R_2-R_3](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D1%262%261%263%5C%5C1%262%26-2%26-3%5C%5C2%26-3%26-1%260%5Cend%7Barray%7D%5Cright%5D%20%5Crightarrow%20R_2-R_1%5Crightarrow%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D1%262%261%263%5C%5C0%260%26-3%26-6%5C%5C2%26-3%26-1%260%5Cend%7Barray%7D%5Cright%5D%5Crightarrow%20%5C%5C%5C%5C%5Crightarrow%20R_3-2R_1%20%5Crightarrow%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D1%262%261%263%5C%5C0%260%26-3%26-6%5C%5C0%26-7%26-3%26-6%5Cend%7Barray%7D%5Cright%5D%20%5Crightarrow%20R_3%5Cleftrightarrow%20R_2%5Crightarrow%20%5C%5C%5C%5C%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D1%262%261%263%5C%5C0%26-7%26-3%26-6%5C%5C0%260%26-3%26-6%5Cend%7Barray%7D%5Cright%5D%5Crightarrow%20R_2-R_3)
![\displaystyle \rightarrow \left[\begin{array}{cccc}1&2&1&3\\0&-7&0&0\\0&0&-3&-6\end{array}\right] \rightarrow R_3\rightarrow -\frac{1}{3}R_3\rightarrow \left[\begin{array}{cccc}1&2&1&3\\0&-7&0&0\\0&0&1&2\end{array}\right]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Crightarrow%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D1%262%261%263%5C%5C0%26-7%260%260%5C%5C0%260%26-3%26-6%5Cend%7Barray%7D%5Cright%5D%20%5Crightarrow%20R_3%5Crightarrow%20-%5Cfrac%7B1%7D%7B3%7DR_3%5Crightarrow%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D1%262%261%263%5C%5C0%26-7%260%260%5C%5C0%260%261%262%5Cend%7Barray%7D%5Cright%5D)
Отсюда:
![\displaystyle \begin{cases}x+2y+z=3\\-7y=0\\z=2\end{cases} \Rightarrow \begin{cases}x=1\\y=0\\z=2\end{cases}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cbegin%7Bcases%7Dx%2B2y%2Bz%3D3%5C%5C-7y%3D0%5C%5Cz%3D2%5Cend%7Bcases%7D%20%5CRightarrow%20%5Cbegin%7Bcases%7Dx%3D1%5C%5Cy%3D0%5C%5Cz%3D2%5Cend%7Bcases%7D)
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