Solving Systems by Gauss-Jordan Elimination I'm going through my textbook solving the practice problems, I haven't had any trouble solving systems that are already in row-echelon form, or reduced row-echelon form. However, I'm struggling with using the Gaussian and Gauss-Jordan methods to get them to this point. One of the questions I have is:
$$x_1+x_2+2x_3=8$$
$$-x_1-2x_2+3x_3=1$$
$$3x_1-7x_2+4x_3=10$$
Which as an augmented matrix would be:
$$\begin{align}
\begin{bmatrix}
1 & 2 & 3 & 8\\
-1 & -2 & 3 & 1\\
3 & -7 & 4 & 10
\end{bmatrix}
\end{align}
$$
I understand the first step, which is to take a multiple of 1, that when added would set -1 to 0, which in this case would be 1. And this would give me:
$$\begin{align}
\begin{bmatrix}
1 & 2 & 3 & 8\\
0 & -1 & 4 & 2\\
3 & -7 & 4 & 10
\end{bmatrix}
\end{align}
$$
But after this I'm lost as to what I should do next, or if I even did the first step properly. Can any show me how to solve using this method?

Follow the following thirteen steps,  Rx (like R1, R2 or R3) refers to the matrix row number:


*

*Swap R1 with R3

*Add R1/3 to R2

*Multiply R2 by -3/13

*Subtract R1/3 from R3

*Multiply (3/2)R3

*Swap R2 with R3

*Subtract R2/5 from R3

*Multiply (-5/6)R3

*Subtract R3 from R2

*Subtract (4) R3 from R1

*Add (7/5) R2 to R1

*Divide R1 by 3

*Divide R2 by 5


You should arrive at a RREF of:
$$
\left[
\begin{array}{@{}ccc|c@{}}1 & 0 & 0 & 3\\0 & 1 & 0 & 1\\0 & 0 & 1 & 2\end{array}\right]
$$
Now, you read the solution from the bottom up, so we have:


*

*$x_3 = 2$

*$x_2 = 1$

*$x_1 = 3$


You should of course check my work and verify that those value satisfy all three simultaneous equations! 
