Solve the simultaneous equation x - 2y + z = -2 x + y - 2z = 4 -2x + y + z = 12?

x - 2y + z = -2 …… [1]
x + y - 2z = 4 …… [2]
-2x + y + z = 12 …… [3]

[1]*2 :
2x - 4y + 2z = -4 ….. [1']

[2]*2 :
2x + 2y - 4z = 8 …… [2']

[3] + [1'] :
(-2x + 2x) + (y - 4y) + (z + 2z) = 12 - 4
-3y + 3z = 8
3y - 3z = 8 …… [4]

[3] + [2'] :
(-2x + 2x) + (y + 2y) + (z - 4z) = 12 + 8
3y - 3z = 20 …… [5]

[4] and [5] are contradictory.
Hence, the equation system has no solution.

x - 2y + z = -2
x + y - 2z = 4
-2x + y + z = 12
3y - 3z = 6
2x - 4y + 2z = -4
-3y + 3z = 8
No solutions exist.

no solution

x -2y +z = -2 -------(1)
x +y -2z = 4 ----------(2)
-2x+ y + z = 12 ------(3)

multiply equation (2) by -1
-x -y + 2z = -4
x -2y + z = -2 -------(1)
add:

-3y+3z = -6 ----------(4)

multiply equation (2) by 2
2x +2y - 4z = 8
-2x+y + z = 12 --------(3)
add:

3y -3z = 20 -------(5)

Solve (4) & (5)

-3y+3z = -6 ----(4)
3y-3z = 0 ------(5)
add:
0y+0z = -6

This system has no solution
(Inconsistent)

Solve for X, Y,and Z in the following systems of three equations:

a. X + 2Y + Z = 6

X + Y =4

3X + Y + Z = 8

b. 10X + Y + Z = 12

8X + 2Y +Z =11

20X - 10Y -2Z = 8

c. 22X + 5Y + 7Z = 12

10X + 3Y + 2Z =5

9X + 2Y + 12Z = 14