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${:[x_(1)+x_(1)+2x_(3)=-1],[x_(1)-2x_(2)+x_(3)=-5],[3x_(1)+x_(2)+x_(3)=3]:} Find all solutions by using the Guassian elimination and Gauss-Jordan Reduction$

$\begin{array}{l} x_{1}+x_{1}+2 x_{3}=-1 \\ x_{1}-2 x_{2}+x_{3}=-5 \\ 3 x_{1}+x_{2}+x_{3}=3 \end{array}$ $\newline$ Find all solutions by using the Guassian $\newline$ elimination and Gauss-Jordan Reduction

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\begin{array}{l} x_{1}+x_{1}+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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Find all solutions by using the Guassian

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elimination and Gauss-Jordan Reduction

Write Augmented Matrix: Write down the augmented matrix for the system of equations. $\newline$ $\begin{bmatrix} 1 & 0 & 2 & | & -1 \\ 1 & -2 & 1 & | & -5 \\ 3 & 1 & 1 & | & 3 \end{bmatrix}$
Perform Gaussian Elimination Step $1$ : Perform the first step of Gaussian elimination: Make the first element of the first column a $1$ (already done) and use it to make all elements below it in the same column $0$ . $\newline$ Subtract the first row from the second row: $\newline$ $\begin{bmatrix} 1 & 0 & 2 & | & -1 \\ 0 & -2 & -1 & | & -4 \\ 3 & 1 & 1 & | & 3 \end{bmatrix}$ $\newline$ Subtract $3$ times the first row from the third row: $\newline$ $\begin{bmatrix} 1 & 0 & 2 & | & -1 \\ 0 & -2 & -1 & | & -4 \\ 0 & 1 & -5 & | & 6 \end{bmatrix}$
Make Second Element $1$ : Next, make the second element of the second column a $1$ by dividing the second row by $-2$ . $\newline$ $\begin{bmatrix} 1 & 0 & 2 & | & -1 \\ 0 & 1 & 0.5 & | & 2 \\ 0 & 1 & -5 & | & 6 \end{bmatrix}$ $\newline$ Then, use the second row to make all elements above and below it in the same column $0$ . $\newline$ Add the second row to the third row: $\newline$ $\begin{bmatrix} 1 & 0 & 2 & | & -1 \\ 0 & 1 & 0.5 & | & 2 \\ 0 & 0 & -4.5 & | & 8 \end{bmatrix}$
Make Third Element $1$ : Finally, make the third element of the third column a $1$ by dividing the third row by $-4$ . $5$ . $\newline$ $\begin{bmatrix} 1 & 0 & 2 & | & -1 \\ 0 & 1 & 0.5 & | & 2 \\ 0 & 0 & 1 & | & -1.777\ldots \end{bmatrix}$ $\newline$ Then, use the third row to make all elements above it in the same column $0$ . $\newline$ Subtract $2$ times the third row from the first row and subtract $0$ . $5$ times the third row from the second row: $\newline$ $\begin{bmatrix} 1 & 0 & 0 & | & 2.555\ldots \\ 0 & 1 & 0 & | & 3.111\ldots \\ 0 & 0 & 1 & | & -1.777\ldots \end{bmatrix}$