# Write the system of linear equations in the form Ax = b and solve this matrix equation for x. begin{cases}x_1+x_2-3x_3=-1-x_1+2x_2=1x_1-x_2+x_3=2end{cases}

Write the system of linear equations in the form Ax = b and solve this matrix equation for x.
$\left\{\begin{array}{l}{x}_{1}+{x}_{2}-3{x}_{3}=-1\\ -{x}_{1}+2{x}_{2}=1\\ {x}_{1}-{x}_{2}+{x}_{3}=2\end{array}$
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Clara Reese
Firstly, switch to matrix form.
$\left[\begin{array}{ccc}1& 1& -3\\ -1& 2& 0\\ 1& -1& 1\end{array}\right]\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {x}_{3}\end{array}\right]=\left[\begin{array}{c}-1\\ 1\\ 2\end{array}\right]$
Now, form augmented matrix and by Gaussian climination reach row reduced echelon form.
Form augmented matrix $\left[\begin{array}{cccc}1& 1& -3& -1\\ -1& 2& 0& 1\\ 1& -1& 1& 2\end{array}\right]$
$\left[\begin{array}{cccc}1& 1& -3& -1\\ 0& 3& -3& 0\\ 0& -2& 4& 3\end{array}\right]$

From here, we have solution.
${x}_{1}=2\phantom{\rule{0ex}{0ex}}{x}_{2}=1.5\phantom{\rule{0ex}{0ex}}{x}_{3}=1.5$
Result: ${x}_{1}=2\phantom{\rule{0ex}{0ex}}{x}_{2}=1.5\phantom{\rule{0ex}{0ex}}{x}_{3}=1.5$