Solve the given system of equations System of equations is:
 x1 + 2x2 - 3x3 = 9
2x1 -  x2 +  x3 = 0
4x1 -  x2 +  x3 = 4

I start by creating the augmented matrix for this system:
1   2 -3  | 9
2  -1  1  | 0 
4  -1  1  | 4

I then want to reduce this to row-echelon form. I must have missed an easier path, because the numbers were not so pretty. I'll list out the steps, and then give the final result. 
R2 = R2 -  2R1
R3 = R3 -  4R1
R2 = -1/5R2
R3 = -1/9R3
R3 = R3 - R2
R3 = -45/2R2

Reduced-echelon form:
1   2   -3   | 9
0   1   -7/5 | -18/5
0   0   1    | -161

Using the reduced matrix and back substitution, we get our variables
x3 = -161 
x2 = -229
x1 = -16

Here's the problem, when I substitute these numbers into the equations they all check out except for the middle one:
2x1 - x2 + x3 = 0
2(-16) + 229 - 161 = 36
36 != 0

Where did I go wrong? Also, if anyone see's an obviously better way to reduce the matrix please feel free to share. I seem to always struggle with these by missing the easier "moves". 

Simpler method -
eq$3$ - eq$2$ gives $2x_1=4 ;  x_1 =2$
from eq$3$ , $x_2-x_3=4 \implies 2x_2-2x_3=8$
from eq$1$ , $2x_2-3x_3=7$
from these two, $x_3=1$ and then $x_2 =5$
