So the way we go about this is called Gaussian elimination. Not that it matters, its row operations. The very first thing we have to do is make the topleft number a 1. It already is a 1, so yay us. Now what we do is basically a bunch of eliminations. Normally when we have a system of equations, like:
2x + 3y = 6
-x + y = 2
we would multiply the bottom row by 2 and we'd get x's to cancel and go from there. This is exactly what we're going to do. 2 rows at a time, we do a series of eliminations until the numbers we have remaining fit the form above. Now there is a bit of an order we have to do this in. We HAVE to use the first row in each elimination until we get the all the numbers in the left column like
1
0
0
0
So we start with row 1 and row 2
1 2 1 1
2 -3 -1 6
So now we must do elimination like we normally do and our goal is for that bottom left 2 to become 0. We can do this by doing the operation
-2Row 1 + Row 2 (usually just written r sub 1, r sub 2, etc). So -2 R1 + R2 is
-2 -4 -2 -2
2 -3 -1 6
---------------
0 -7 -3 4
Now this result becomes our NEW row 2. We rip old the old row two and put this in its place. Row 1 DOES NOT change. We will keep row 1 like
1 2 1 1, only the row that we made 0 retains its new form. So this is quite a bit, so see if this makes sense first @Data_LG2