Solve the following systems of linear equations, showing all work: 2x+y-z = 14 x+2y-z = 12 2x-2y+z = 3

The given simultaneous equations are:

2x + y - z = 14   ...  (1)

x + 2y - z = 12   ...  (2)

2x - 2y + z = 3   ...  (3)

Adding equation (2) and (3) we get:

x + 2x + 2y - 2y - z + z = 12 + 3

3x = 15

Therefore:

x = 15/3 = 5

Substituting this value of x in equations (1) we get:

10 + y - z = 14

y - z = 14 - 10 = 4 ...  (4)

Substituting this value of x in equations (3) we get:

10 - 2y + z = 3

- 2y + z = 3 - 10 = - 7  ...  (5)

Adding equations (4) and (5) we get:

y - 2y - z + z = 4 - 7

- y = - 3

y = 3

When we substitute value of x and y in equations (1), (2) and (3) we get different values of z.

Substituting values of x and y in equation (1) we get:

10 + 3 - z = 14

- z = 14 - 10 - 3  = 1

Substituting values of x and y in equation (2) we get:

5 + 6 - z = 12

-z = 1

z = -1

Substituting values of x and y in equation (3) we get:

10 - 6 + z = 3

z = 3 - 10 + 6 = -1

As we get different value of zwhen substituting in different e equation, thegiven equations are not consistent with each other. Thus there no unique values of x, y, and z can be determined using these three equations as simultaneous equations.