{:[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 using Gaussian elimination or Gauss-Jordan reduction.

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Write Matrix Form:  Write down the system of equations in matrix form: $$ \begin{bmatrix} 1 & 1 & 2 & | & -1 \ 1 & -2 & 1 & | & -5 \ 3 & 1 & 1 & | & 3 \end{bmatrix} $$Gaussian Elimination:  Begin Gaussian elimination by making the first element of the first column a 1 (if it's not already) and use it to zero out the rest of the first column: - Subtract the first row from the second row. - Subtract 3 times the first row from the third row. $$ \begin{bmatrix} 1 & 1 & 2 & | & -1 \ 0 & -3 & -1 & | & -4 \ 0 & -2 & -5 & | & 6 \end{bmatrix} $$Make Second Element 1:  Make the second element of the second column a 1 by dividing the second row by -3: $$ \begin{bmatrix} 1 & 1 & 2 & | & -1 \ 0 & 1 & 1/3 & | & 4/3 \ 0 & -2 & -5 & | & 6 \end{bmatrix} $$Zero Out Second Column:  Use the second row to zero out the rest of the second column: - Add the second row to the first row. - Add 2 times the second row to the third row. $$ \begin{bmatrix} 1 & 0 & 7/3 & | & -1/3 \ 0 & 1 & 1/3 & | & 4/3 \ 0 & 0 & -13/3 & | & 14/3 \end{bmatrix} $$Make Third Element 1:  Make the third element of the third column a 1 by dividing the third row by -13/3: $$ \begin{bmatrix} 1 & 0 & 7/3 & | & -1/3 \ 0 & 1 & 1/3 & | & 4/3 \ 0 & 0 & 1 & | & -14/13 \end{bmatrix} $$Zero Out Third Column:  Use the third row to zero out the rest of the third column: - Subtract 7/3 times the third row from the first row. - Subtract 1/3 times the third row from the second row. $$ \begin{bmatrix} 1 & 0 & 0 & | & 3 \ 0 & 1 & 0 & | & 5 \ 0 & 0 & 1 & | & -14/13 \end{bmatrix} $$

$\begin{array}{l} x_{1}+x_{2}+2 x_{3}=-1 \\ x_{1}-2 x_{2}+x_{3}=-5 \\ 3 x_{1}+x_{2}+x_{3}=3 \end{array}$ $\newline$ a)Find all solutions using Gaussian elimination or Gauss-Jordan reduction.

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x1+x2+2x3=−1x1−2x2+x3=−53x1+x2+x3=3 \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} x1​+x2​+2x3​=−1x1​−2x2​+x3​=−53x1​+x2​+x3​=3​\newlinea)Find all solutions using Gaussian elimination or 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 using Gaussian elimination or Gauss-Jordan reduction.