<s>[INST] Solve a system of three equations by rewriting in row-echelon form I tried solving this system of equations and I got what seems to be an inconsistent system.  I wanted to post my results here to see if I'm correct.  
Here is the original problem:
$$ \left\{ \begin{aligned} 3x_1-2x_2+4x_3&=1 \\ x_1+x_2-2x_3&=3 \\ 2x_1-3x_2+6x_3&=8 \end{aligned} \right. $$
For the first step I interchanged equation one with equation two to give the following:
$$ \left\{ \begin{aligned} x_1+x_2-2x_3&=3 \\ 3x_1-2x_2+4x_3&=1 \\ 2x_1-3x_2+6x_3&=8 \end{aligned} \right. $$
I then added (-3) times equation one to equation two to give a new equation two:
$$ \left\{ \begin{aligned} x_1+x_2-2x_3&=3 \\ -5x_2+10x_3&=-8 \\ 2x_1-3x_2+6x_3&=8 \end{aligned} \right. $$
Next I added (-2) times equation one to equation three to give a new equation three:
$$ \left\{ \begin{aligned} x_1+x_2-2x_3&=3 \\ -5x_2+10x_3&=-8 \\ -5x_2+10x_3&=-2 \end{aligned} \right. $$
So it was at this point that I thought something was wrong, but I kept going.  Next I added (-1) times equation two to equation three to give a new equation three:
$$ \left\{ \begin{aligned} x_1+x_2-2x_3&=3 \\ -5x_2+10x_3&=-8 \\ 0=10 \end{aligned} \right. $$
Did I do something wrong, or is the properly solved?  We haven't gotten to matrices yet, which I know will be tremendously easier, our teacher wants us to get comfortable with doing it this way first.  
 [/INST] I redid the calculation on the computer, it yields:
$$\begin{bmatrix} 3 & -2 & 4 & 1 \\ 1 & 1 & -2 & 3 \\ 2 & -3 & 6 & 
8 \\ \end{bmatrix}
\xrightarrow{R_1 \leftrightarrow R_2}
\begin{bmatrix} 1 & 1 & -2 & 3 \\ 3 & -2 & 4 & 1 \\ 2 & -3 & 6 & 
8 \\ \end{bmatrix}
\xrightarrow{R_2 \gets R_2 -3 R_1}
\begin{bmatrix} 1 & 1 & -2 & 3 \\ 0 & -5 & 10 & -8 \\ 2 & -3 & 6 & 
8 \\ \end{bmatrix}
\xrightarrow{R_3 \gets R_3 -2 R_1}
\begin{bmatrix} 1 & 1 & -2 & 3 \\ 0 & -5 & 10 & -8 \\ 0 & -5 & 10 & 
2 \\ \end{bmatrix}
\xrightarrow{R_3 \gets R_3 -R_2}
\begin{bmatrix} 1 & 1 & -2 & 3 \\ 0 & -5 & 10 & -8 \\ 0 & 0 & 0 & 
10 \\ \end{bmatrix}$$
which matches your final answer.
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