Gaussian Elimination by Bianca Avila

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.

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