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${:[x_(1)+x_(2)+2x_(3)=-1],[x_(1)-2x_(2)+x_(3)=-5],[3x_(1)+x_(2)+x_(3)=3]:} a)Find all solutions using Gaussian elimination or Gauss-Jordan reduction.$

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

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Grade 8

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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 using Gaussian elimination or Gauss-Jordan reduction.

Write Matrix Form: Write down the system of equations in matrix form: $\newline$ $\begin{bmatrix} 1 & 1 & 2 & | & -1 \\ 1 & -2 & 1 & | & -5 \\ 3 & 1 & 1 & | & 3 \end{bmatrix}$
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: $\newline$ - Subtract the first row from the second row. $\newline$ - Subtract $3$ 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 Element $1$ : Make the second element of the second column a $1$ by dividing the second row by $-3$ : $\newline$ $\begin{bmatrix} 1 & 1 & 2 & | & -1 \\ 0 & 1 & 1/3 & | & 4/3 \\ 0 & -2 & -5 & | & 6 \end{bmatrix}$
Zero Out Second Column: Use the second row to zero out the rest of the second column: $\newline$ - Add the second row to the first row. $\newline$ - Add $2$ times the second row to the third row. $\newline$ $\begin{bmatrix} 1 & 0 & 7/3 & | & -1/3 \\ 0 & 1 & 1/3 & | & 4/3 \\ 0 & 0 & -13/3 & | & 14/3 \end{bmatrix}$
Make Third Element $1$ : Make the third element of the third column a $1$ by dividing the third row by $-13$ / $3$ : $\newline$ $\begin{bmatrix} 1 & 0 & 7/3 & | & -1/3 \\ 0 & 1 & 1/3 & | & 4/3 \\ 0 & 0 & 1 & | & -14/13 \end{bmatrix}$
Zero Out Third Column: Use the third row to zero out the rest of the third column: $\newline$ - Subtract $7$ / $3$ times the third row from the first row. $\newline$ - Subtract $1$ / $3$ times the third row from the second row. $\newline$ $\begin{bmatrix} 1 & 0 & 0 & | & 3 \\ 0 & 1 & 0 & | & 5 \\ 0 & 0 & 1 & | & -14/13 \end{bmatrix}$