Okay, prepare yourself for a huge reply. Also, it seems my pretty tabs and spacing have been killed by the formatting on this, so I hope it'll be clear enough for you to see which columns lie where.
Normally you'd use Gauss-Jordan elimination to reduce a matrix to reduced-row-echelon form, but this one has too many variables. In any case, the augmented matrix would look like this:
23070
305-2619
-1-2-3-4-5
(Not going to bother with the matrix lines on the sides, meh)
I'd normally start reducing by moving the row with the 1 at the first entry to the top. You could start it any other way you want, though, like dividing the first row by 2 or something. But I don't like fractions, so I'm doing it this way!
So after poking Row 3 up to the first row, it would look like this:
-1-2-3-4-5
23070
305-2619
Then multiply the current Row 1 with -1 to get:
12345
23070
305-2619
Take Row 1, multiply it by -2 and add this answer to Row 2 to get rid of the 2 in Column 1:
12345
0-1-6-1-10
305-2619
Now you want Row 2 Column 2 to have a positive 1 instead of a negative, so multiply the whole Row 2 with -1:
12345
016110
305-2619
You also want to get rid of that annoying 3 in Row 3 Column 1, so multiply Row 1 with -3 and add this to Row 3.
12345
016110
0-6-4-384
You now want to get Column 2 to have only zeros, except for Row 2. So take Row 2, multiply by -2 and add this to Row 1. At the same time (just to save time/space/effort), also multiply Row 2 by 6 and add this answer to Row 3.
10-92-15
016110
0032-3264
We're finally getting close to the end. We want Row 3 Column 3 to be 1, so multiply the entire row by 1/32.
10-92-15
016110
001-12
Now to get Column 3, Rows 1 and 2 to be 0. Multiply Row 3 by -6 and 9 and add to their respective columns.
100-73
0107-2
001-12
And there you are. It might be wise to just go over all my calculations and make sure there aren't any mistakes with the adding or multiplication. I hope it helped, and I hope you'll be able to do it on your own from now on
It's really easy, it just takes an age to finish the stupid thing.
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