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${:[x_(1)+x_(2)+2x_(3)=-1],[x_(1)-2x_(2)+x_(3)=-5],[3x_(1)+x_(2)+x_(3)=3]:} a) Find all solutions by uning the Gaussian elimination & Gauss-Jordan Reduction$

$\begin{array}{l} x_{1}+x_{2}+2 x_{3}=-1 \\ x_{1}-2 x_{2}+x_{3}=-5 \\ 3 x_{1}+x_{2}+x_{3}=3 \end{array}$ $\newline$ a) Find all solutions by uning the Gaussian elimination \& Gauss-Jordan Reduction

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

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a) Find all solutions by uning the Gaussian elimination \& Gauss-Jordan Reduction

Set up augmented matrix: Set up the augmented matrix for the system of equations: $\newline$ $\begin{bmatrix} 1 & 1 & 2 & | & -1 \\ 1 & -2 & 1 & | & -5 \\ 3 & 1 & 1 & | & 3 \end{bmatrix}$
Perform Gaussian elimination: Perform the first step of Gaussian elimination to make zeros below the pivot in the first column: $\newline$ Subtract the first row from the second row and subtract three times the first row from the third row: $\newline$ $\begin{bmatrix} 1 & 1 & 2 & | & -1 \\ 0 & -3 & -1 & | & -4 \\ 0 & -2 & -5 & | & 6 \end{bmatrix}$
Make second pivot to $1$ : Next, make the second pivot (second row, second column) to $1$ by dividing the second row by $-3$ : $\newline$ $\begin{bmatrix} 1 & 1 & 2 & | & -1 \\ 0 & 1 & 1/3 & | & 4/3 \\ 0 & -2 & -5 & | & 6 \end{bmatrix}$
Eliminate second column: Eliminate the second column below and above the second pivot: $\newline$ Add two times the second row to the third row and subtract the second row from the first row: $\newline$ $\begin{bmatrix} 1 & 0 & 5/3 & | & -7/3 \\ 0 & 1 & 1/3 & | & 4/3 \\ 0 & 0 & -13/3 & | & 14/3 \end{bmatrix}$
Make third pivot to $1$ : Make the third pivot (third row, third column) to $1$ by dividing the third row 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 third column: Eliminate the third column above the third pivot: $\newline$ Subtract $5$ / $3$ times the third row from the first row and subtract $1$ / $3$ times the third row from the second row: $\newline$ $\begin{bmatrix} 1 & 0 & 0 & | & 1 \\ 0 & 1 & 0 & | & 2 \\ 0 & 0 & 1 & | & -14/13 \end{bmatrix}$