We begin with this system of equations:

3x+7y-4z = -46

5w+4x+8y+z = 7

8w+4y-2z = 0

-1w+6x+2z = 13

The corresponding augmented matrix is:

0 3 7 -4 -46

5 4 8 1 7

8 0 4 -2 0

-1 6 0 2 13

We need a pivot in column one but the 0 in position (1,1) cannot be a pivot

**Step 1**: Permute R1 and R3

8 0 4 -2 0

5 4 8 1 7

0 3 7 -4 -46

-1 6 0 2 13

**Step 2**: Turn (1,1) into a 1

R1 = R1 * 1/8

1 0 1/2 -1/4 0

5 4 8 1 7

0 3 7 -4 -46

-1 6 0 2 13

now we have a pivot in column 1, so we need to turn all numbers underneath it into a zero

**Step 3**: R2 = (-5*R1) + R2

1 0 1/2 -1/4 0

0 4 11/2 9/4 7

0 3 7 -4 -46

-1 6 0 2 13

**Step 4**: R4 = R1 + R4

1 0 1/2 -1/4 0

0 4 11/2 9/4 7

0 3 7 -4 -46

0 6 1/2 7/4 13

now we need a pivot in column 2

**Step 5**: R2 = R2 * 1/4

1 0 1/2 -1/4 0

0 1 11/8 9/16 7/4

0 3 7 -4 -46

0 6 1/2 7/4 13

again change all numbers underneath the pivot into a zero

**Step 6**: R3 = (-3*R2) + R3

1 0 1/2 -1/4 0

0 1 11/8 9/16 7/4

0 0 23/8 -91/16 -205/4

0 6 1/2 7/4 13

**Step 7**: R4 = (-6*R2) + R3

1 0 1/2 -1/4 0

0 1 11/8 9/16 7/4

0 0 23/8 -91/16 -205/4

0 0 -31/4 -13/8 5/2

now we need a pivot in column 3

**Step 8**: R3 = R3 * 8/23

1 0 1/2 -1/4 0

0 1 11/8 9/16 7/4

0 0 1 -91/46 -410/23

0 0 -31/4 -13/8 5/2

again change all numbers under the pivot into a zero

**Step 9**: R4 = (31/4 * R3) + R4

1 0 1/2 -1/4 0

0 1 11/8 9/16 7/4

0 0 1 -91/46 -410/23

0 0 0 -390/23 -3120/23

**Step 10**: Obtain a pivot in column 4

R4 = R4 * -23/390

1 0 1/2 -1/4 0

0 1 11/8 9/16 7/4

0 0 1 -91/46 -410/23

0 0 0 1 8

**Step 11**: Now we work our way upwards to change everything above a pivot into a zero

R3 = (R4 * 91/46) + R3

1 0 1/2 -1/4 0

0 1 11/8 9/16 7/4

0 0 1 0 -2

0 0 0 1 8

**Step 12**: R2 = (R4 * -9/16) + R2

1 0 1/2 -1/4 0

0 1 11/8 0 -11/4

0 0 1 0 -2

0 0 0 1 8

**Step 13**: R1 = (R4 * 1/4) + R1

1 0 1/2 0 2

0 1 11/8 0 -11/4

0 0 1 0 -2

0 0 0 1 8

**Step 14**: R2 = (R3 * -11/8) + R2

1 0 1/2 0 2

0 1 0 0 0

0 0 1 0 -2

0 0 0 1 8

Almost done…

**Step 15**: R1 = (R3 * -1/2) + R1

1 0 0 0 3

0 1 0 0 0

0 0 1 0 -2

0 0 0 1 8

This matrix now gives us the solutions to the system of equations we began with.

**Solutions**:

w = 3

x = 0

y = -2

z = 8

You can plug the values back into the equations to verify the solutions are correct.