Answer
$\begin{cases}x_1=-4+2x_2 \\ x_2 \text{ is free}\\ x_3=-7 \end{cases}$
Work Step by Step
To find the general solution, we use row operations on the augmented matrix $\left[
\begin{array}{cccc}
1 & -2 & -1 & 3\\
3 & -6 & -2 & 2
\end{array}
\right]$ to obtain an equivalent matrix in reduced row echelon form. Then we solve the corresponding system of equations.
First, we want to eliminate the 3 in the second row since it is below the leading term of the first row. To do this, we use row-replacement. So we replace row 2 with -3*(row 1)+(row 2), which gives
$$\left[
\begin{array}{cccc}
1 & -2 & -1 & 3\\
0 & 0 & 1 & -7
\end{array}
\right].$$
Next, we need to eliminate the -1 above the leading one of the second row. We will again use row-replacement. We replace row 1 with (row 2) + (row 1) to get
$$\left[
\begin{array}{cccc}
1 & -2 & 0 & -4\\
0 & 0 & 1 & -7
\end{array}
\right].$$
Now we solve the corresponding system of equations, which is
$$x_1-2x_2=-4\\ x_3=-7.$$
From the reduced row echelon form of the matrix, we see that $x_1$ and $x_3$ are the leading variables (since they correspond to the pivot positions of the matrix) and $x_2$ is free. So we want to solve for $x_1$ in terms of $x_2$. Thus our system becomes
$$x_1=-4+2x_2\\ x_3=-7, $$
and, hence our general solution is $$\begin{cases}x_1=-4+2x_2 \\ x_2 \text{ is free}\\ x_3=-7 \end{cases}.$$
