How to 'Row Reduce' this matrix

Can someone please help me solve this row reduction?
I need to figure out what steps to apply to solving this

2 -2 -3 : 1
5 2 1: 1
-1 5 4: 1

Thank you

2 -2 -3 : 1
5 2 1: 1
-1 5 4: 1

this a 3X3 system with x, y, z
it's the same as:

2x - 2y - 3z = 1
5x + 2y + z = 1
- x + 5y + 4z = 1

when we have it in it's 'reduced row echelon form':
1 0 0 : x
0 1 0 : y
0 0 1 : z
you have it solved for some x, y, and z

2 -2 -3 : 1
5 2 1: 1
-1 5 4: 1

you have many options to get it into that form.

let R1 be row 1, etc...
change R3 to (R1 + 2R3)
2 -2 -3 : 1
5 2 1: 1
0 8 5: 3

change R1 to (R2 - 2R1)
1 6 7 : -1
5 2 1 : 1
0 8 5 : 3

change R2 to (R2 - 5R5)
1 6 7 : -1
0 -28 -34: 6
0 8 5 : 3

change R2 to (-1/2)R2

1 6 7 : -1
0 14 17: -3
0 8 5 : 3

notice in these last two matrices the second and third row (by themselves) are a 2X2 system...

change R3 to 4R2 - 7R3 (the lcm of 14 and 8 is 56)

1 6 7 : -1
0 14 17: -3
0 0 33 : -33

change R3 to (1/33)R3

1 6 7 : -1
0 14 17: -3
0 0 1 : -1

change R1 to R1 - 7R3
and
change R2 to R2 - 17R3

1 6 : -1