Help with this system.?

8x + 9y + 6z = 27 ...... [1]
3x - 5y + 8z = 76 ...... [2]
6x - 3y - 6z = -75 ...... [3]

[3] × (4/3) :
8x - 4y - 8z = -100 ...... [4]

[1] - [4]:
13y + 14z = 127 ...... [5]

[2] × 2:
6x - 10y + 16z = 152 ...... [6]

[3] - [6]:
7y - 22z = -227
y = (22z - 227)/7 ....... [7]

Sub. [7] into [5] :
[13(22z - 227)/7] + 14z = 127
(286z/7) - (2951/7) + 14z = 127
286z - 2951 + 98z = 889
384z = 3840
z = 10

Sub. z = 10 into [7] :
y = (22 * 10 - 227)/7
y = -1

Sub. y = -1 and z = 10 into [2] :
3x - 5 * (-1) + 8 * 10 =76
3x + 85 = 76
3x = -9
x = -3

Then, x = -3, y = -1, z = 10

(1) : 8x + 9y + 6z = 27

(2) : 3x - 5y + 8z = 76

(3) : 6x - 3y - 6z = - 75


You calculate (3) - [2 * (2)] and you can get the equation (4)

(4) : (6x - 3y - 6z) - 2.(3x - 5y + 8z) = - 75 - (2 * 76)

(4) : 6x - 3y - 6z - 6x + 10y - 16z = - 75 - 152

(4) : 7y - 22z = - 227

(4) : 7y = 22z - 227

(4) : y = (22z - 227)/7


You calculate [3 * (1)] - [8 * (2)]

3.(8x + 9y + 6z) - 8.(3x - 5y + 8z) = (3 * 27) - (8 * 76)

24x + 27y + 18z - 24x + 40y - 64z = 81 - 608

67y - 46z = - 527 → recall (4): y = (22z - 227)/7

67.[(22z - 227)/7] - 46z = - 527

67.(22z - 227) - 322z = - 3689

1474z - 15209 - 322z = - 3689

1152z = 11520

→ z = 10


Recall (4): y = (22z - 227)/7

y = (220 - 227)/7

→ y = - 1


Recall (2): 3x - 5y + 8z = 76

3x = 76 + 5y - 8z

3x = 76 - 5 - 80

3x = - 9

→ x = - 3

I do not understand.