# Use row operations to solve this system x + y - z = -8 4x - y + z = 8 x - 3y + 2z = 5

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### 2 Answers

x + y - z = -8 (1)

4x - y + z = 8 (2)

x - 3y + 2z = 5 (3)

Adding (1) + (2):

5x=0, then **x=0** (4)

Multiply (1) x 2:

2x + 2y -2z = -16 (5)

Add (5) + (3):

3x - y =-11, **y=11** (6)

Substitute (4) and (6) in (1):

z=11+8=19, **z=19**

**x=0, y=11, z=19**

Verify by replacing in (3):

0 - 3*11 +2*19 = 5

The given system of equations can be expressed as a matrix as:

`[[1, 1, -1, -8],[4, -1, 1, 8],[1, -3, 2, 5]]`

Add -4 times the first row to the 2nd row.

`[[1, 1, -1, -8],[0, -5, 5, 40],[1, -3, 2, 5]]`

Add -1 times the 1st row to the third.

`[[1, 1, -1, -8],[0, -5, 5, 40],[0, -4, 3, 13]]`

Multiply the 2nd row by -1/5.

`[[1, 1, -1, -8],[0, 1, -1, -8],[0, -4, 3, 13]]`

Add 4 times the 2nd row to the 3rd.

`[[1, 1, -1, -8],[0, 1, -1, -8],[0, 0, -1, -19]]`

Multiply the third row by -1

`[[1, 1, -1, -8],[0, 1, -1, -8],[0, 0, 1, 19]]`

Add the 3rd row to the 2nd.

`[[1, 1, -1, -8],[0, 1, 0, 11],[0, 0, 1, 19]]`

Add the 3rd row to the 1st

`[[1, 1, 0, 11],[0, 1, 0, 11],[0, 0, 1, 19]]`

Subtract the 2nd row from the 1st

`[[1, 0, 0, 0],[0, 1, 0, 11],[0, 0, 1, 19]]`

**The solution of the system of equations is x = 0, y = 11 and z = 19**