How to solve systems of equations in three variables?

2x - 2y + 4z = -3 ...... [1]
4x + 2y - 2z = -1 ...... [2]
6x - 4y - 2z = -18 ...... [3]

[1] + [2] :
6x + 2z = -4
2z = -6x - 4 ...... [4]

[2] - [3] :
-2x + 6y = 17
6y = 2x + 17
2y = (2x + 17)/3 ...... [5]

Substitute [4] and [5] into [2] :
4x + [(2x + 17)/3] - (-6x - 4) = -1
3{4x + [(2x + 17)/3] - (-6x - 4)} = 3(-1)
12x + 2x + 17 + 18x + 12 = -3
32x = -32
x = -1

Substitute x = -1 into [5] :
2y = [2(-1) + 17]/3
y = 2.5

Substitute x = -1 into [4] :
2z = -6(-1) - 4
z = 1

Ans: x = -1, y = 2.5, z = 1

2x - 2y + 4z = -3
4x + 2y - 2z = -1
6x - 4y - 2z = -18

You can use Cramer's rule. Calcuate the following determinants.

|2. -2. 4|
|4. 2. -2| = D = -128
|6 -4. -2|

|-3. -2. 4|
|-1. 2. -2| = D_x = 128
|-18 -4 -2|

|2. -3. 4|
|4. -1. -2| = D_y = -320
|6 -18 -2|

|2. -2. -3|
|4. 2. -1| = D_z = -128
|6.-4 -18|

x = D_x/D = 128/(-128) = -1
y = D_y/D = -320/(-128) = 5/2
z = D_z/D = -128/(-128) = 1

6x - 4y - 2z = -18
2(3x - 2y - z) = -18
3x - 2y - z = -9
3x - 2y + 9 = z

2x - 2y + 4(3x - 2y + 9) = -3
2x - 2y + 12x - 8y + 36 = -3
14x - 10y = -39

4x + 2y - 2(3x - 2y + 9) = -1
4x + 2y - 6x + 4y - 18 = -1
-2x + 6y = 17

14x - 10y = -39
-2x + 6y = 17

14x - 10y = -39
-14x + 42y = 119
32y = 80
y = 2.5

-2x + 6 * 2.5 = 17
-2x + 15 = 17
-2x = 2
x = -1

3 * -1 - 2 * 2.5 + 9 = z
-3 - 5 + 9 = z
1 = z

x = -1
y = 2.5
z = 1