math help please EXPLAIN?

x+y+z=6 -----(1)
x-y+2z=5 ----(2)
x-y+3z=8 ----(3)

(1)+(2)
2x+3z = 11 ------(4)
(1)+(3)
2x+4z = 14 ------(5)

(4)-(5)
-z = -3
z = 3 (z solved)
2x+3z = 11 ------(4)
2x+3(3) = 11
2x+9 = 11
2x=2
x= 1 ( x solved)

x+y+z = 6 ----(1)
1+y+3 = 6
y=2 (y solved)

x=1; y=2; z=3

x + y + z = 6 ...... [1]
x - y + 2z = 5 ...... [2]
x - y + 3z = 8 ...... [3]

[1] - [2] :
2y - z = 1 ...... [4]

[1] - [3] :
2y - 2z = -2 ...... [5]

[4] - [5] :
z = 3

Put z = 3 into [4] :
2y - (3) = 1
2y = 4
y = 2

Put z = 3 and y = 2 into [1] :
x + (2) + (3) = 6
x = 1

Then, x = 1, y = 2, z = 3

x + y + z = 6
x - y + 2z = 5
x - y + 3z = 8
2x + 3z = 11
z = 3
x = 1
y = 2
Solution:
x = 1, y = 2, z = 3