# 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}

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

2020-11-10
Firstly, switch to matrix form.
$$\begin{bmatrix}1&1&-3\\-1&2&0\\1&-1&1\end{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}=\begin{bmatrix}-1\\1\\2\end{bmatrix}$$
Now, form augmented matrix and by Gaussian climination reach row reduced echelon form.
Form augmented matrix $$\begin{bmatrix}1&1&-3&-1\\-1&2&0&1\\1&-1&1&2\end{bmatrix}$$
$$R_2+R_1\rightarrow\ and\ R_3-R_1\rightarrow R_3$$ $$\begin{bmatrix}1&1&-3&-1\\0&3&-3&0\\0&-2&4&3\end{bmatrix}$$
$$R_2/3\rightarrow R_2\ \ \begin{bmatrix}1&1&-3&-1\\0&3&-3&0\\0&-2&4&3\end{bmatrix}$$
$$R_1-R_2\rightarrow R_2\ and\ R_3+2R_2\rightarrow R_3\ and\ R_3/3\rightarrow R_3 \begin{bmatrix}1&0&-2&-1\\0&1&0&1.5\\0&0&1&1.5\end{bmatrix}$$
$$R_1+2R_3\rightarrow R_1\ and\ R_2+R_3\rightarrow R_2\ \begin{bmatrix}1&0&0&2\\0&1&0&1.5\\0&0&1&1.5\end{bmatrix}$$
From here, we have solution.
$$x_1=2\\x_2=1.5\\x_3=1.5$$
Result: $$x_1=2\\x_2=1.5\\x_3=1.5$$

### Relevant Questions

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