Question

решите систему методом гаусса​

Answer 1

$\displaystyle\begin{cases}x_1-2x_2+3x_3-x_4=9\\4x_1+3x_2-x_3+2x_4=5\\2x_1-5x_2+3x_3+x_4=16\\4x_1+6x_2+2x_3-x_4=5\end{cases}$

$\displaystyle\left[\begin{array}{cccccccc}1&-2&3&-1&|&9&|&\downarrow\\4&3&-1&2&|&5&|&*(-4)\\2&-5&3&1&|&16&|&*(-2)\\4&6&2&-1&|&5&|&*(-4)\end{array}\right]\to\\\to\left[\begin{array}{cccccccc}1&-2&3&-1&|&9&|&\\0&11&-13&6&|&-31&|&\downarrow\\0&-1&-3&3&|&-2&|&\uparrow (*(-1))\\0&14&-10&3&|&-31&|&\end{array}\right]\to\\\to\left[\begin{array}{cccccccc}1&-2&3&-1&|&9&|&\\0&1&3&-3&|&2&|&\downarrow\\0&11&-13&6&|&-31&|&*(-11)\\0&14&-10&3&|&-31&|&*(-14)\end{array}\right]\to$

$\to\left[\begin{array}{cccccccc}1&-2&3&-1&|&9&|&\\0&1&3&-3&|&2&|&\\0&0&-46&39&|&-53&|&:(-46)\\0&0&-52&45&|&-59&|&\end{array}\right]\to\\\to\left[\begin{array}{cccccccc}1&-2&3&-1&|&9&|&\\0&1&3&-3&|&2&|&\\0&0&1&-\frac{39}{46}&|&\frac{53}{46}&|&\downarrow\\0&0&-52&45&|&-59&|&*52\end{array}\right]\to$

$\to\left[\begin{array}{cccccccc}1&-2&3&-1&|&9&|&\\0&1&3&-3&|&2&|&\\0&0&1&-\frac{39}{46}&|&\frac{53}{46}&|&\\0&0&0&\frac{42}{46}&|&\frac{42}{46}&|&*\frac{46}{42}\end{array}\right]\to\\\to\left[\begin{array}{cccccccc}1&-2&3&-1&|&9&|&\\0&1&3&-3&|&2&|&\\0&0&1&-\frac{39}{46}&|&\frac{53}{46}&|&\\0&0&0&1&|&1&|&\end{array}\right]$

$\begin{cases}x_1-2x_2+3x_3-x_4=9\to x_1=2\\x_2+3x_3-3x_4=2\to x_2=-1\\x_3-\frac{39}{46}x_4=\frac{53}{46}\to x_3=2\\x_4=1\end{cases}\\\left[\begin{array}{c}x_1\\x_2\\x_3\\x_4\end{array}\right]\to \left[\begin{array}{c}2\\-1\\2\\1\end{array}\right]$