Problem:

Using only elementary row operations, solve the following system of linear equations:

{3x - y + 5z + 4w = -11}

{-2x + y -3z -3w = 8}

My steps: I keep going in circles. I can't seem to get rid of enough variables to make a dent in this.

R1 +R2 -> R1 {x + 2z + w = -3}

{-2x + y -3 z - 3w = 8}

R2 + 2R1 -> R2 {x +2z +w = -3}

{y + z - w = 2}

R1 - 2R2 -> R1 {x - 2y + 3w = -1}

{y + z - w = 2}

2R2 -> R2 {x - 2y + 3w = -1}

{2y + 2z + 2w = 4}

R1 + R2 -> R1 {x + 2z + w = 3}

{2y + 2z - 2w = 4}

R2 - R1 -> R2 {x + 2z + w = 3}

{-x + 2y = 1}

R1 + R2 -> R2 {x + 2z + w = 3}

{2y + 2z + w = 2}

Do you see how I'm going in circles? Am I even taking the correct first step?