Пошаговое объяснение:
1) Метод Гаусса:
2) Метод Крамера
0\\\\\Delta_1=\begin{vmatrix}4&2&1\\1&-5&3\\8&7&-1\end{vmatrix}=4\cdot\begin{vmatrix}-5&3\\7&-1\end{vmatrix}-2\cdot\begin{vmatrix}1&3\\8&-1\end{vmatrix}+1\cdot\begin{vmatrix}1&-5\\8&7\end{vmatrix}=33\\\\\Delta_2=\begin{vmatrix}1&4&1\\3&1&3\\2&8&-1\end{vmatrix}=1\cdot\begin{vmatrix}1&3\\8&-1\end{vmatrix}-4\cdot\begin{vmatrix}3&3\\2&-1\end{vmatrix}+1\cdot\begin{vmatrix}3&1\\2&8\end{vmatrix}=33\\\\\Delta_3=\begin{vmatrix}1&2&4\\3&-5&1\\2&7&8\end{vmatrix}=1\cdot\begin{vmatrix}-5&1\\7&8\end{vmatrix}-2\cdot\begin{vmatrix}3&1\\2&8\end{vmatrix}+4\cdot\begin{vmatrix}3&-5\\2&7\end{vmatrix}=33\\\\x_1=x_2=x_3={\Delta_1\over\Delta}={\Delta_2\over\Delta}={\Delta_3\over\Delta}={33\over33}=1" alt="\displaystyle\Large\left( \begin{array}{*{3}{c}|c}1&2&1&4\\3&-5&3&1\\2&7&-1&8\end{array}\right)\\\\\Delta=\begin{vmatrix}1&2&1\\3&-5&3\\2&7&-1\end{vmatrix}=1\cdot\begin{vmatrix}-5&3\\7&-1\end{vmatrix}-2\cdot\begin{vmatrix}3&3\\2&-1\end{vmatrix}+1\cdot\begin{vmatrix}3&-5\\2&7\end{vmatrix}=33>0\\\\\Delta_1=\begin{vmatrix}4&2&1\\1&-5&3\\8&7&-1\end{vmatrix}=4\cdot\begin{vmatrix}-5&3\\7&-1\end{vmatrix}-2\cdot\begin{vmatrix}1&3\\8&-1\end{vmatrix}+1\cdot\begin{vmatrix}1&-5\\8&7\end{vmatrix}=33\\\\\Delta_2=\begin{vmatrix}1&4&1\\3&1&3\\2&8&-1\end{vmatrix}=1\cdot\begin{vmatrix}1&3\\8&-1\end{vmatrix}-4\cdot\begin{vmatrix}3&3\\2&-1\end{vmatrix}+1\cdot\begin{vmatrix}3&1\\2&8\end{vmatrix}=33\\\\\Delta_3=\begin{vmatrix}1&2&4\\3&-5&1\\2&7&8\end{vmatrix}=1\cdot\begin{vmatrix}-5&1\\7&8\end{vmatrix}-2\cdot\begin{vmatrix}3&1\\2&8\end{vmatrix}+4\cdot\begin{vmatrix}3&-5\\2&7\end{vmatrix}=33\\\\x_1=x_2=x_3={\Delta_1\over\Delta}={\Delta_2\over\Delta}={\Delta_3\over\Delta}={33\over33}=1" align="absmiddle" class="latex-formula">
3) Матричный метод
0\\\\A^{T}=\begin{pmatrix}1&3&2\\2&-5&7\\1&3&-1\end{pmatrix}\\\\A_{ij}=(-1)^{i+j}\cdot M_{ij}\\\\A_{1,1}=\begin{vmatrix}-5&7\\3&-1\end{vmatrix}=-16\;A_{1,2}=-\begin{vmatrix}2&7\\1&-1\end{vmatrix}=9\;A_{1,3}=\begin{vmatrix}2&-5\\1&3\end{vmatrix}=11\\\;A_{2,1}=-\begin{vmatrix}3&2\\3&-1\end{vmatrix}=9\;A_{2,2}=\begin{vmatrix}1&2\\1&-1\end{vmatrix}=-3\;A_{2,3}=-\begin{vmatrix}1&3\\1&3\end{vmatrix}=0\\\;A_{3,1}=\begin{vmatrix}3&2\\-5&7\end{vmatrix}=31\;A_{3,2}=-\begin{vmatrix}1&2\\2&7\end{vmatrix}=-3\;A_{3,3}=\begin{vmatrix}1&3\\2&-5\end{vmatrix}=-11\\\\A^{-1}={1\over33}\begin{pmatrix}-16&9&11\\9&-3&0\\31&-3&-11\end{pmatrix}\\\\X={1\over33}\begin{pmatrix}-16&9&11\\9&-3&0\\31&-3&-11\end{pmatrix}\begin{pmatrix}4\\1\\8\end{pmatrix}={1\over33}\begin{pmatrix}-16\cdot4+9\cdot1+11\cdot8\\9\cdot4-3\cdot1+0\cdot8\\31\cdot4-3\cdot1-11\cdot8\end{pmatrix}=}{1\over33}\begin{pmatrix}33\\33\\33\end{pmatrix}=\begin{pmatrix}1\\1\\1\end{pmatrix}" alt="\displaystyle\large A\cdot X=B\Rightarrow X=A^{-1}\cdot B\\\\A=\begin{pmatrix}1&2&1\\3&-5&3\\2&7&-1\end{pmatrix},B=\begin{pmatrix}4\\1\\8\end{pmatrix},X=\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}\\\\\\\Delta=\begin{vmatrix}1&2&1\\3&-5&3\\2&7&-1\end{vmatrix}=1\cdot\begin{vmatrix}-5&3\\7&-1\end{vmatrix}-2\cdot\begin{vmatrix}3&3\\2&-1\end{vmatrix}+1\cdot\begin{vmatrix}3&-5\\2&7\end{vmatrix}=33>0\\\\A^{T}=\begin{pmatrix}1&3&2\\2&-5&7\\1&3&-1\end{pmatrix}\\\\A_{ij}=(-1)^{i+j}\cdot M_{ij}\\\\A_{1,1}=\begin{vmatrix}-5&7\\3&-1\end{vmatrix}=-16\;A_{1,2}=-\begin{vmatrix}2&7\\1&-1\end{vmatrix}=9\;A_{1,3}=\begin{vmatrix}2&-5\\1&3\end{vmatrix}=11\\\;A_{2,1}=-\begin{vmatrix}3&2\\3&-1\end{vmatrix}=9\;A_{2,2}=\begin{vmatrix}1&2\\1&-1\end{vmatrix}=-3\;A_{2,3}=-\begin{vmatrix}1&3\\1&3\end{vmatrix}=0\\\;A_{3,1}=\begin{vmatrix}3&2\\-5&7\end{vmatrix}=31\;A_{3,2}=-\begin{vmatrix}1&2\\2&7\end{vmatrix}=-3\;A_{3,3}=\begin{vmatrix}1&3\\2&-5\end{vmatrix}=-11\\\\A^{-1}={1\over33}\begin{pmatrix}-16&9&11\\9&-3&0\\31&-3&-11\end{pmatrix}\\\\X={1\over33}\begin{pmatrix}-16&9&11\\9&-3&0\\31&-3&-11\end{pmatrix}\begin{pmatrix}4\\1\\8\end{pmatrix}={1\over33}\begin{pmatrix}-16\cdot4+9\cdot1+11\cdot8\\9\cdot4-3\cdot1+0\cdot8\\31\cdot4-3\cdot1-11\cdot8\end{pmatrix}=}{1\over33}\begin{pmatrix}33\\33\\33\end{pmatrix}=\begin{pmatrix}1\\1\\1\end{pmatrix}" align="absmiddle" class="latex-formula">