solving system of equation using matrices

I have another matirx question. I understand the concepts but am just not seeing it all the way through. This question is killing me!

Question: Solve the system of equations using matrices;

. . ...w +..x +..y +..z =..5
. . ...w + 2x -..y - 2z = -1
. . ...w - 3x - 3y -. .z = -1
. . .2w -. .x + 2y -..z = -2

First I created the matrix:

. . .[ 1..1. .1..1 |..5 ]
. . .[ 1..2 -1 -2 | -1 ]
. . .[ 1 -3 -3 -1 | -1 ]
. . .[ 2 -1..2 -1 | -2 ]

Then I performed the operations:

Row 1 = OK, leave alone.
Row 2 = R2 - R1
Row 3 = R3 - R1
Row 4 = R4 - 2R1

After which I got:

. . .[..1..1..1.. 1 |. .5 ]
. . .[..0..1 -2 -3 |..-6 ]
. . .[..0 -4 -4 -2 |..-6 ]
. . .[..0 -3..0 -3 | -12 ]

Next I did:

Row 1 = OK, leave alone.
Row 2 = OK, leave alone.
Row 3 = R3 + 4R2
Row 4 = R4 + 3R2

To get:

. . .[ 1 1. ..1. ..1 |. ..5 ]
. . .[ 0 1. .-2. .-3 |. .-6 ]
. . .[ 0 0 -12 -14 | -30 ]
. . .[ 0 0. .-6 -12 | -30 ]

Now I'm stuck!!! AM I correct to this point? If so, how do I get -12 to 1? I tried:

Row 3 = R3/-12
Row 4 = 1/2R3 + R4

Which gave me:

. . .[ 1 1..1. ..1 | . .5 ]
. . .[ 0 1 -2. .-3 |..-6 ]
. . .[ 0 0..1 7/6 | 5/2 ]
. . .[ 0 0..0. .-5 | -15 ]

After all the back substition I ended up with:

. . .w = -22/7
. . .x = 31/7
. . .y = 5/7
. . .z = 3

But this doesn't work after I plugged in the original equations to check.

Any help would be GREATLY appreciated.

Thank you!

At this point:

. . .[ 1. .1. ..1. . .1 |. ..5 ]
. . .[ 0. .1..-2. ..-3 |. .-6 ]
. . .[ 0. .0 -12 -14 | -30 ]
. . .[ 0. .0..-6..-12 | -30 ]

I'd divide row 4 by -6, and switch the result with row 3:

. . .[ 1. .1. ..1. ..1 | . .5 ]
. . .[ 0. .1..-2. .-3 |. .-6 ]
. . .[ 0. .0. ..1. ..2 | . .5 ]
. . .[ 0. .0 -12 -14 | -30 ]

Now, multiply R3 by 12, add it to R4, and put the result in row 4:

. . .[ 1. .1. .1. .1 | . .5 ]
. . .[ 0. .1..-2..-3 |..-6 ]
. . .[ 0. .0. .1. ..2 |. .5 ]
. . .[ 0. .0. .0..10 | 30 ]

Ok...take it from here. I got a solution which checked in all of the original equations.

Thanks Mrspi,

I got it! {(2, 1, -1, 3)}

I had been scratching my head on that one for an hour.

Thank you.