Question

olve the system of equations using matrices. Use Gaussian elimination with back-substitution.
x + y + z     =      -5
x - y + 3z    =     -1
4x + y + z    =     -2

I need it explained to me, how do I do this?

Answer 1

you need to add/subtract rows and their multiples so the resulting matrix consists of a diagonal of 1's.

To start, subtract R1 from R2:
x + y + z = -5
0 - 2y + 2z = 4
4x + y + z = -2

Now subtract 4*R1 from R3:

x + y + z = -5
0 - 2y + 2z = 4
0 - 3y - 3z = 18

Now work on the other rows. When the left side is just a diagonal of 1's, the right side will be the solution, because it will look something like

x + 0 + 0 = 12
0 + y + 0 = 4
0 + 0 + z = -3