{:[x_(1)+x_(2)+2x_(3)=-1],[x_(1)-2x_(2)+x_(3)=-5],[3x_(1)+x_(2)+x_(3)=3]:} find all solutions by veing the Gaussian elimination l Lzwss - Jordan Reduction.

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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]:}  find all solutions by veing the Gaussian elimination l Lzwss - Jordan Reduction.

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Write Augmented Matrix:  Write the augmented matrix for the system of equations. $$ \begin{bmatrix} 1 & 1 & 2 & | & -1 \ 1 & -2 & 1 & | & -5 \ 3 & 1 & 1 & | & 3 \end{bmatrix} $$Leading 1 in R1:  Perform row operations to get a leading 1 in the first row, first column (R1). No changes needed as the first element is already 1.Zeros in First Column:  Make zeros below the leading 1 in the first column using R2 - R1 → R2 and R3 - 3R1 → R3. $$ \begin{bmatrix} 1 & 1 & 2 & | & -1 \ 0 & -3 & -1 & | & -4 \ 0 & -2 & -5 & | & 6 \end{bmatrix} $$Leading 1 in R2:  Make the second column's second row element into a leading 1 by multiplying R2 by -1/3. $$ \begin{bmatrix} 1 & 1 & 2 & | & -1 \ 0 & 1 & 1/3 & | & 4/3 \ 0 & -2 & -5 & | & 6 \end{bmatrix} $$Eliminate First Column:  Eliminate the first column's second row element by adding R2 to R1 and add 2R2 to R3. $$ \begin{bmatrix} 1 & 0 & 7/3 & | & 1/3 \ 0 & 1 & 1/3 & | & 4/3 \ 0 & 0 & -13/3 & | & 14/3 \end{bmatrix} $$Leading 1 in R3:  Convert the third row's third column element to a leading 1 by multiplying R3 by -3/13. $$ \begin{bmatrix} 1 & 0 & 7/3 & | & 1/3 \ 0 & 1 & 1/3 & | & 4/3 \ 0 & 0 & 1 & | & -14/13 \end{bmatrix} $$Eliminate Third Column:  Eliminate the third column's first and second row elements by subtracting (7/3)R3 from R1 and subtracting (1/3)R3 from R2. $$ \begin{bmatrix} 1 & 0 & 0 & | & 29/39 \ 0 & 1 & 0 & | & 50/39 \ 0 & 0 & 1 & | & -14/13 \end{bmatrix} $$Check Solution:  Check the solution by substituting back into the original equations. Substituting $ x_1 = 29/39, x_2 = 50/39, x_3 = -14/13 $ into the original equations confirms the solution is correct.

$x_{1}+x_{2}+2x_{3}=-1$ $\newline$ $x_{1}-2x_{2}+x_{3}=-5$ $\newline$ $3x_{1}+x_{2}+x_{3}=3$ $\newline$ find all solutions by using the Gaussian elimination or Gauss-Jordan Reduction.

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x1+x2+2x3=−1x_{1}+x_{2}+2x_{3}=-1x1​+x2​+2x3​=−1\newlinex1−2x2+x3=−5x_{1}-2x_{2}+x_{3}=-5x1​−2x2​+x3​=−5\newline3x1+x2+x3=33x_{1}+x_{2}+x_{3}=33x1​+x2​+x3​=3\newline find all solutions by using the Gaussian elimination or Gauss-Jordan Reduction.

$x_{1}+x_{2}+2x_{3}=-1$ $\newline$ $x_{1}-2x_{2}+x_{3}=-5$ $\newline$ $3x_{1}+x_{2}+x_{3}=3$ $\newline$ find all solutions by using the Gaussian elimination or Gauss-Jordan Reduction.