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Problem 10: Solve the following systems of linear equations by Gaussian elimination method

 Problem 10: Solve the following systems of linear equations by Gaussian elimination method:

\begin{align*} 2x-2y+3z=2\\ x+2y-z=3\\ 3x-y+2z=1 \end{align*}

Solution:

Consider the augmented matrix of this system and apply row operations.

\[\left[\begin{array}{c|c} A & B \end{array} \right] = \begin{align*} \left[\begin{array}{rrr|r} 2 & -2 & 3& 2 \\ 1 & 2& -1 & 3 \\ 3 & -1 & 2 & 1 \\ \end{array}\right] \xrightarrow{R_1\leftrightarrow R_2} \left[\begin{array}{rrr|r} 1 & 2& -1 & 3 \\ 2 & -2 & 3& 2 \\ 3 & -1 & 2 & 1 \\ \end{array}\right] \end{align*}\] \[\xrightarrow{R2\rightarrow R_2-2R_1,R3\rightarrow R_3-3R_1} \left[\begin{array}{rrr|r} 1 & 2& -1 & 3 \\ 0 & -6 & 5& -4 \\ 0 & -7 & 5 & -8 \\ \end{array}\right] \xrightarrow{R3\rightarrow R_3-R_2} \left[\begin{array}{rrr|r} 1 & 2& -1 & 3 \\ 0 & -6 & 5& -4 \\ 0 & -1 & 0 & -4 \\ \end{array}\right]\] \[\xrightarrow{R_3\rightarrow -R_3} \left[\begin{array}{rrr|r} 1 & 2& -1 & 3 \\ 0 & -6 & 5& -4 \\ 0 & 1 & 0 & 4 \\ \end{array}\right]\]

For Gaussian elimination we write down the equations corresponding to REF. Then we get

\begin{align*} x+2y+-z=3\\ -6y+5z=-4\\ y=4 \end{align*}

By solving we get

Solution set S is \begin{align*} y=4\\ z=4\\ x=-1 \end{align*}

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