Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

${:[x_(1)+x_(2)+2x_(3)=-1],[x_(1)-2x_(2)+x_(3)=-5],[3x_(1)+x_(2)+x_(3)=3]:} a) Find all solutions by wising the Gaussian elimination & Gaus-Jordan Reduction$

$\begin{array}{l} x_{1}+x_{2}+2 x_{3}=-1 \\ x_{1}-2 x_{2}+x_{3}=-5 \\ 3 x_{1}+x_{2}+x_{3}=3 \end{array}$ $\newline$ a) Find all solutions by wising the Gaussian elimination \& Gaus-Jordan Reduction

View step-by-step help

Home

Math Problems

Grade 8

Write an equation word problem

Full solution

\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 by wising the Gaussian elimination \& Gaus-Jordan Reduction

Write Augmented Matrix: Write the augmented matrix for the system of equations. $\newline$ $\begin{bmatrix} 1 & 1 & 2 & | & -1 \\ 1 & -2 & 1 & | & -5 \\ 3 & 1 & 1 & | & 3 \end{bmatrix}$
First Gaussian Elimination: Perform the first step of Gaussian elimination: Make the first element of the first column a $1$ (already done) and use it to zero out the rest of the first column. $\newline$ Subtract the first row from the second row: $\newline$ $\begin{bmatrix} 1 & 1 & 2 & | & -1 \\ 0 & -3 & -1 & | & -4 \\ 3 & 1 & 1 & | & 3 \end{bmatrix}$ $\newline$ Subtract $3$ times the first row from the third row: $\newline$ $\begin{bmatrix} 1 & 1 & 2 & | & -1 \\ 0 & -3 & -1 & | & -4 \\ 0 & -2 & -5 & | & 6 \end{bmatrix}$
Second Gaussian Elimination: Make the second element of the second column a $1$ by dividing the second row by $-3$ . $\newline$ $\begin{bmatrix} 1 & 1 & 2 & | & -1 \\ 0 & 1 & 1/3 & | & 4/3 \\ 0 & -2 & -5 & | & 6 \end{bmatrix}$ $\newline$ Use the second row to zero out the rest of the second column. $\newline$ Add $2$ times the second row to the third row: $\newline$ $\begin{bmatrix} 1 & 1 & 2 & | & -1 \\ 0 & 1 & 1/3 & | & 4/3 \\ 0 & 0 & -13/3 & | & 22/3 \end{bmatrix}$
Third Gaussian Elimination: Make the third element of the third column a $1$ by dividing the third row by $-13$ / $3$ . $\newline$ $\begin{bmatrix} 1 & 1 & 2 & | & -1 \\ 0 & 1 & 1/3 & | & 4/3 \\ 0 & 0 & 1 & | & -2 \end{bmatrix}$ $\newline$ Use the third row to zero out the rest of the third column. $\newline$ Subtract $2$ times the third row from the first row and subtract $1$ / $3$ times the third row from the second row: $\newline$ $\begin{bmatrix} 1 & 1 & 0 & | & 3 \\ 0 & 1 & 0 & | & 2 \\ 0 & 0 & 1 & | & -2 \end{bmatrix}$
Final Gaussian Elimination: Use the second row to zero out the rest of the second column. $\newline$ Subtract the second row from the first row: $\newline$ $\begin{bmatrix} 1 & 0 & 0 & | & 1 \\ 0 & 1 & 0 & | & 2 \\ 0 & 0 & 1 & | & -2 \end{bmatrix}$