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question: $\begin{cases} x_{1}+x_{2}+2x_{3}=-1 \ x_{1}-2x_{2}+x_{3}=-5 \ 3x_{1}+x_{2}+x_{3}=3 \end{cases}$ a) Find all solutions by using the Gaussian elimination & Gaus-Jordan Reduction

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Write an equation word problem

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Q. question:

\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}

a) Find all solutions by using the Gaussian elimination & Gaus-Jordan Reduction

Write Equations: Step $1$ : Write down the system of equations. $\newline$ $\begin{align*} x_1 + x_2 + 2x_3 &= -1 \\ x_1 - 2x_2 + x_3 &= -5 \\ 3x_1 + x_2 + x_3 &= 3 \end{align*}$
Convert to Matrix: Step $2$ : Convert the system into an augmented matrix. $\newline$ $\begin{bmatrix} 1 & 1 & 2 & |-1 \\ 1 & -2 & 1 & |-5 \\ 3 & 1 & 1 & |3 \end{bmatrix}$
Leading $1$ in R $1$ : Step $3$ : Perform row operations to get a leading $1$ in the first row, first column ( $R1$ is already set). $\newline$ No changes needed for $R1$ .
Make Column Zero: Step $4$ : Make the elements below the leading $1$ in the first column zero. $\newline$ Subtract R $1$ from R $2$ and subtract $3$ times R $1$ from R $3$ . $\newline$ $\begin{bmatrix} 1 & 1 & 2 & |-1 \\ 0 & -3 & -1 & |-4 \\ 0 & -2 & -5 & |6 \end{bmatrix}$
Leading $1$ in R $2$ : Step $5$ : Get a leading $1$ in the second row, second column. $\newline$ Divide R $2$ by $-3$ . $\newline$ $\begin{bmatrix} 1 & 1 & 2 & |-1 \\ 0 & 1 & 1/3 & |4/3 \\ 0 & -2 & -5 & |6 \end{bmatrix}$
Make Column Zero: Step $6$ : Make the elements above and below the leading $1$ in the second column zero. $\newline$ Add $-1$ times R $2$ to R $1$ and add $2$ times R $2$ to 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 R $3$ : Step $7$ : Get a leading $1$ in the third row, third column. $\newline$ Divide 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 Column Zero: Step $8$ : Make the elements above the leading $1$ in the third column zero. $\newline$ Subtract $5$ / $3$ times R $3$ from R $1$ and subtract $1$ / $3$ times R $3$ from R $2$ . $\newline$ $\begin{bmatrix} 1 & 0 & 0 & |-1 \\ 0 & 1 & 0 & |2 \\ 0 & 0 & 1 & |-14/13 \end{bmatrix}$

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