To solve for this let's first set up the matrix with the columns being the coefficients of x1, x2, and x3 and the fourth column being what each equation equals (which is -9 in this case).
2 -1 3 -9
-13 -2 1 -9
-4 -5 2 -9
Alright, so to start out, we want to get a 1 in the top left corner and zeros below it. I'm going to do this dividing the first row by 2, then adding 13 times R1 and 4 times R1 to R2 and R3 respectively. This generates the following matrix.
1 -.5 1.5 -4.5
0 -8.5 20.5 -67.5
0 -7 8 -27
Next you'll want to get a 1 in that middle value of the second column and 1 below it. I'd do this by dividing by -8.5 and then adding 7 times R2 to R3
1 -.5 1.5 -4.5
0 1 -41/17 135/17
0 0 -151/17 486/17
Finally get that last value in the third column to be a 1. We can simply divide the row by -151/17, getting that row to be 0 0 1 -486/151. This means that x3 = -486/151. From there, Gaussian elimination is complete, and we move onto the Jordan elimination. To do this, we'll add multiples of the last row to the first two rows so that there are zeros above that 1 in the third column. That'll look like this.
1 -.5 0 -2817/151
0 1 0 459/2567
0 0 1 -486/151
And lastly, we'll just add 1/2 of R2 to R1 to get the identity matrix in those first 3 columns.
1 0 0 2169/5134
0 1 0 459/2567
0 0 1 -486/151
And now you're done, and you've solved for your 3 variables.
x1 = 2169/5134
x2 = 459/2567
x3 = -486/151
