Solve the system using Gaussian elimination and back-substitution. Help me please... this is all confusing

Question

 -x + 3y −  z = -9

  x − 2y + 2z = 8

 2x −  y +  z = 7

Answer 1

Hello Lucy!

Let's first write the augmented matrix:

-1 3 -1 | -9

1 -2 2 | 8

2 -1 1 | 7

You have to do row operations until you get to the echelon form. Let's call the first row R1 and so on. The first operation I see is R1 + R2, to zero the first coefficient in R2. We get:

-1 3 -1 | -9

0 1 1 | -1

2 -1 1 | 7

The following operation is 2R1 + R3, to zero the first coefficient in R3. We get

-1 3 -1 | -9

0 1 1 | -1

0 5 -1 | -11

Now we do -3R2 + R1 to zero the second coefficient in R1. We get:

-1 0 -4 | -6

0 1 1 | -1

0 5 -1 | -11

Now we do -5R2 + R3 to zero the second coefficient in R3. We get:

-1 0 -4 | -6

0 1 1 | -1

0 0 -6 | -6

Now we do -1/6 * R3

-1 0 -4 | -6

0 1 1 | -1

0 0 1 | 1

If you are paying attention we just found that z = 1. Now we do -R3+R2 :

-1 0 -4 | -6

0 1 0 | -2

0 0 1 | 1

we just found that y = -2. Now we do 4R3+R1

-1 0 0 | -2

0 1 0 | -2

0 0 1 | 1

And to finish it, we do -R1

1 0 0 | 2

0 1 0 | -2

0 0 1 | 1

We just found that x = 2. The solving our system.