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${:[x_(1)+x_(2)^(')+2x_(3)=-1],[x_(1)-2x_(2)+x_(3)=-5],[3x_(1)+x_(2)+x_(3)=3]:} find all solutions by waing the Gausian elimination 2 Gauss-Jordan$

$\begin{array}{l} x_{1}+x_{2}^{\prime}+2 x_{3}=-1 \\ x_{1}-2 x_{2}+x_{3}=-5 \\ 3 x_{1}+x_{2}+x_{3}=3 \end{array}$ $\newline$ find all solutions by waing the Gausian elimination $2$ Gauss-Jordan

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\begin{array}{l} x_{1}+x_{2}^{\prime}+2 x_{3}=-1 \\ x_{1}-2 x_{2}+x_{3}=-5 \\ 3 x_{1}+x_{2}+x_{3}=3 \end{array}

\newline

find all solutions by waing the Gausian elimination

2

Gauss-Jordan

Write Augmented Matrix: Write the augmented matrix for the system of equations. $\newline$ $\begin{bmatrix} 1 & 1 & 2 & | & -1 \\ 1 & -2 & 1 & | & -5 \\ 3 & 1 & 1 & | & 3 \end{bmatrix}$
Leading $1$ in R $1$ : Perform row operations to get a leading $1$ in the first row, first column ( $R1$ is already set). $\newline$ No changes needed for $R1$ .
Eliminate Below Leading $1$ : Make the elements below the leading $1$ in the first column zero, using R $2$ - R $1$ → R $2$ and $3$ R $1$ - R $3$ → R $3$ . $\newline$ $\begin{bmatrix} 1 & 1 & 2 & | & -1 \\ 0 & -3 & -1 & | & -4 \\ 0 & -2 & -5 & | & 6 \end{bmatrix}$
Leading $1$ in R $2$ : Get a leading $1$ in the second row, second column by dividing R $2$ by $-3$ . $\newline$ $\begin{bmatrix} 1 & 1 & 2 & | & -1 \\ 0 & 1 & 1/3 & | & 4/3 \\ 0 & -2 & -5 & | & 6 \end{bmatrix}$
Eliminate Above and Below R $2$ : Eliminate the second column below and above the leading $1$ in R $2$ . Add $2$ R $2$ to R $3$ and subtract R $2$ from R $1$ . $\newline$ $\begin{bmatrix} 1 & 0 & 5/3 & | & -7/3 \\ 0 & 1 & 1/3 & | & 4/3 \\ 0 & 0 & -13/3 & | & 14/3 \end{bmatrix}$
Leading $1$ in R $3$ : Get a leading $1$ in the third row, third column by dividing R $3$ by $-13$ / $3$ . $\newline$ $\begin{bmatrix} 1 & 0 & 5/3 & | & -7/3 \\ 0 & 1 & 1/3 & | & 4/3 \\ 0 & 0 & 1 & | & -14/13 \end{bmatrix}$
Eliminate Above R $3$ : Eliminate the third column above R $3$ by subtracting ( $5$ / $3$ )R $3$ from R $1$ and subtracting ( $1$ / $3$ )R $3$ from R $2$ . $\newline$ $\begin{bmatrix} 1 & 0 & 0 & | & 1 \\ 0 & 1 & 0 & | & 2 \\ 0 & 0 & 1 & | & -14/13 \end{bmatrix}$