# Use matrices to solve the system of equations (if possible). Use gaussian elimination with back-substitution. 3x - 2y + z = 15 -x + y + 2z = -10 x - y - 4z = 14

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### 1 Answer

The set of equations

3x - 2y + z = 15

-x + y + 2z = -10

x - y - 4z = 14

has to be solved.

This can be written in matrix form as:

`[(3, -2, 1),(-1, 1, 2), (1, -1, -4)][(x), (y), (z)] = [(15), (-10), (14)]`

In augmented form this is:

`[(3, -2, 1, |15),(-1, 1, 2,|-10), (1, -1, -4,|14)][(x),(y),(z)]`

Add row 2 to row 3

`[(3, -2, 1, |15),(-1, 1, 2,|-10), (0, 0, -2,|4)][(x),(y),(z)]`

Add twice of row 2 to row 1

`[(1, 0, 5, |-5),(-1, 1, 2,|-10), (0, 0, -2,|4)][(x),(y),(z)]`

Add row 1 to row 2

`[(1, 0, 5, |-5),(0, 1, 7,|-15), (0, 0, -2,|4)][(x),(y),(z)]`

Subtract row 3 and row 2 from row 1

`[(1, -1, 0, |6),(0, 1, 7,|-15), (0, 0, -2,|4)][(x),(y),(z)]`

The matrix equation can now be written as:

`[(1, -1, 0),(0, 1, 7), (0, 0, -2)][(x),(y),(z)]=[(6),(-15),(4)]`

This gives z = -2

Back substitution gives y = -1 and x = 5

The solution of the given set of equations is x = 5, y = -1 and z = -2

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