Question

solve the system of equation for all variables using a matix

x-y-z=1
2x-3y+z=10
x+y-2z=0

Answer 1

form the augmented matrix

1 -1 -1 1
2 -3 1 0
1 2 -2 0

=
1 -1 -1 1
0 1 -3 -8 I did 2x#1 - #2
0 -2 1 1 I did #1 - #1

=
1 -1 -1 1
0 1 -3 -8
0 0 -5 -15 I did 2x#2 - #3

=
1 -1 -1 1
0 1 -3 -8
0 0 1 3

At this point you can work backwards since the last line says
z = 3
put that back into line 2, which says
y - 3z = -8
etc

Answer 2

Another way:
If you know how to find the inverse of a 3x3 matrix, find it, then multiply that inverse matrix by the column matrix
1
10
0

here is how to find the inverse of a 3x3 matrix, this is the method I use.

http://www.wikihow.com/Inverse-a-3X3-Matrix