**Gaussian Elimination**

Here is the System of Equations:

x+ y+ z+ w=4

x- 2y- z- w=3

2x- y+ z- w=2

x- y+ 2z- 2w=-7

This is the Augmented Matrix:

1 1 1 1 4

1 -2 -1 -1 3

2 -1 1 -1 2

1 -1 2 -2 -7

**Step1:** We need (2,1) to become 0 so we add (-1*r1) to r2

1 1 1 1 4

0 -3 -2 -2 -1

2 -1 1 -1 2

1 -1 2 -2 -7

**Step 2:** Make (3,1) a 0 we add (-2*r1) to r3

1 1 1 1 4

0 -3 -2 -2 -1

0 -3 -1 -3 -6

1 -1 2 -2 -7

**Step3:** Make (4,1) a 0 we add (-1*r1) to r4

1 1 1 1 4

0 -3 -2 -2 -1

0 -3 -1 -3 -6

0 -2 1 -3 -11

**Step4:** Make (2,2) a pivot by dividing r2 by -3

1 1 1 1 4

0 1 (2/3) (2/3) (1/3)

0 -3 -1 -3 -6

0 -2 1 -3 -11

**Step 5:** Make (3,2) a 0 by adding (3*r2) to r3

1 1 1 1 4

0 1 (2/3) (2/3) (1/3)

0 0 1 -1 -5

0 -2 1 -3 -11

**Step 6:** Make (4,2) a 0 by adding (2*r2) to r4

1 1 1 1 4

0 1 (2/3) (2/3) (1/3)

0 0 1 -1 -5

0 0 (7/3) (-5/3) (-31/3)

**Step 7:** Since (3,3) is a pivot we make (4,3) into a 0 by adding ((-7/3)*r3) to r4

1 1 1 1 4

0 1 (2/3) (2/3) (1/3)

0 0 1 -1 -5

0 0 0 (2/3) (4/3)

**Step 8:**Make (4,4) into a pivot by dividing r4 by (2/3)

1 1 1 1 4

0 1 (2/3) (2/3) (1/3)

0 0 1 -1 -5

0 0 0 1 2

**Step 9:** Make (3,4) to a 0 by adding (1*r4) to r3

1 1 1 1 4

0 1 (2/3) (2/3) (1/3)

0 0 1 0 -3

0 0 0 1 2

**Step 10:** Make (2,4) to a 0 by adding ((-2/3)*r4) to r2

1 1 1 1 4

0 1 (2/3) 0 -1

0 0 1 0 -3

0 0 0 1 2

**Step 11:** Make (1,4) to a 0 by adding (-1*r4) to r1

1 1 1 0 2

0 1 (2/3) 0 -1

0 0 1 0 -3

0 0 0 1 2

**Step 12:** Make (2,3) to a 0 by adding ((-2/3)*r3) to r2

1 1 1 0 2

0 1 0 0 1

0 0 1 0 -3

0 0 0 1 2

**Step 13:** Make (1,3) to a 0 by adding (-1*r3) to r1

1 1 0 0 5

0 1 0 0 1

0 0 1 0 -3

0 0 0 1 2

**Step 14:** Last make (1,2) to a 0 by adding (-1*r2) to r1

1 0 0 0 4

0 1 0 0 1

0 0 1 0-3

0 0 0 1 2

The solutions would be:

x=4

y=1

z=-3

w=2