{:[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 waing the Gausionan elimination?&Gaus-Jordan

Question

Bytelearn · Accepted Answer

Write Augmented Matrix:  Write down the augmented matrix for the system of equations. $$ \begin{bmatrix} 1 & 1 & 2 & | & -1 \ 1 & -2 & 1 & | & -5 \ 3 & 1 & 1 & | & 3 \end{bmatrix} $$Perform Row Operations:  Perform the first row operation: R2 = R2 - R1. $$ \begin{bmatrix} 1 & 1 & 2 & | & -1 \ 0 & -3 & -1 & | & -4 \ 3 & 1 & 1 & | & 3 \end{bmatrix} $$Divide Second Row:  Perform the second row operation: R3 = R3 - 3R1. $$ \begin{bmatrix} 1 & 1 & 2 & | & -1 \ 0 & -3 & -1 & | & -4 \ 0 & -2 & -5 & | & 6 \end{bmatrix} $$Perform Third Row Operation:  Divide the second row by -3 to make the leading coefficient 1. $$ \begin{bmatrix} 1 & 1 & 2 & | & -1 \ 0 & 1 & 1/3 & | & 4/3 \ 0 & -2 & -5 & | & 6 \end{bmatrix} $$Divide Third Row:  Perform the third row operation: R3 = R3 + 2R2. $$ \begin{bmatrix} 1 & 1 & 2 & | & -1 \ 0 & 1 & 1/3 & | & 4/3 \ 0 & 0 & -13/3 & | & 22/3 \end{bmatrix} $$Perform Back Substitution:  Divide the third row by -13/3 to make the leading coefficient 1. $$ \begin{bmatrix} 1 & 1 & 2 & | & -1 \ 0 & 1 & 1/3 & | & 4/3 \ 0 & 0 & 1 & | & -2 \end{bmatrix} $$ Perform back substitution starting from the third row. Solve for $x_3$: $x_3 = -2$. Then, substitute $x_3$ into the second row equation to solve for $x_2$: $x_2 + 1/3(-2) = 4/3$, which simplifies to $x_2 = 2$. Finally, substitute $x_2$ and $x_3$ into the first row equation to solve for $x_1$: $x_1 + 2 + 2(-2) = -1$, which simplifies to $x_1 = 1$.

$\begin{array}{l} x_{1}+x_{2}+2 x_{3}=-1 \\ x_{1}-2 x_{2}+x_{3}=-5 \\ 3 x_{1}+x_{2}+x_{3}=3 \end{array}$ $\newline$ Find all solutions by waing the Gausionan $\newline$ elimination& Gaus-Jordan

Full solution

More problems from Write an equation word problem

Related topics

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​\newlineFind all solutions by waing the Gausionan\newlineelimination& Gaus-Jordan

$\begin{array}{l} x_{1}+x_{2}+2 x_{3}=-1 \\ x_{1}-2 x_{2}+x_{3}=-5 \\ 3 x_{1}+x_{2}+x_{3}=3 \end{array}$ $\newline$ Find all solutions by waing the Gausionan $\newline$ elimination& Gaus-Jordan