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Find all solutions by using the Gaussian elimination & Gauss-Jordan method: $\begin{cases} x_1+x_2+2x_3=-1 \ x_1-2x_2+x_3=-5 \ 3x_1+x_2+x_3=3 \end{cases}$

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Write an equation word problem

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Q. Find all solutions by using the Gaussian elimination & Gauss-Jordan method:

\begin{cases} x_1+x_2+2x_3=-1 \ x_1-2x_2+x_3=-5 \ 3x_1+x_2+x_3=3 \end{cases}

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 First Row: Perform row operations to get a leading $1$ in the first row, first column ( $R_1$ is already set). $\newline$ No changes needed for $R_1$ .
Make Elements Zero: 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 Second Row: Get a leading $1$ in the second row, second column by dividing R $2$ by $-3$ . $\newline$ $\begin{bmatrix} 1 & 1 & 2 & | & -1 \\ 0 & 1 & 1/3 & | & 4/3 \\ 0 & -2 & -5 & | & 6 \end{bmatrix}$
Make Elements Zero: Make the elements above and below the leading $1$ in the second column zero, using R $1$ - R $2$ → R $1$ and R $3$ + $2$ R $2$ → R $3$ . $\newline$ $\begin{bmatrix} 1 & 0 & 5/3 & | & -7/3 \\ 0 & 1 & 1/3 & | & 4/3 \\ 0 & 0 & -13/3 & | & 14/3 \end{bmatrix}$
Leading $1$ in Third Row: Get a leading $1$ in the third row, third column by dividing R $3$ by $-13$ / $3$ . $\newline$ $\begin{bmatrix} 1 & 0 & 5/3 & | & -7/3 \\ 0 & 1 & 1/3 & | & 4/3 \\ 0 & 0 & 1 & | & -14/13 \end{bmatrix}$
Make Elements Zero: Make the elements above the leading $1$ in the third column zero, using R $1$ - $5$ / $3$ R $3$ → R $1$ and R $2$ - $1$ / $3$ R $3$ → R $2$ . $\newline$ $\begin{bmatrix} 1 & 0 & 0 & | & 1 \\ 0 & 1 & 0 & | & 2 \\ 0 & 0 & 1 & | & -14/13 \end{bmatrix}$

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