add the first two equations term for term ...
2x + 2z = 8
x + z = 4
substitute 4 in for (x + z) in the second equation ...
4 - y = 2
y = 2
add the second and third equations ...
2x + y = 4
sub in 2 for y ...
2x + 2 = 4
x = 1
sub x = 1 and y = 2 into the first equation ...
1 + 2 + z = 6
z = 3
Skeeter went to a great deal of work to show that the three planes intersect at the single point (1, 2, 3) but apparently you did not ask that.
You have already been shown how to determine that x + y +z =6 is the unique plane containing the points (6, 0, 0), (0, 6, 0), and (0, 0, 6).
x - y + z =2
If x= y= 0, z= 2. If x= z= 0, -y= 2, y= -2. If y= z= 0, x= 2.
x- y+ z= 2 is the unique plane containing the points (0, 0, 2), (0, -2, 0), and (2, 0, 0).
x + 2y - z = 2
if x= y= 0, -z= 2 so z=-2. If x= z= 0, 2y= 2 so y= 1. If y= z= 0, x= 2.
x+ 2y- z= 2 is the unique plane containing the points (0, 0, -2), (0, 1, 0), and (2, 0, 0).
You have been shown how to draw the plane x + y +z = 6 and you have been shown how to solve the equations algebraically. I suggest you use a Computer Algebra System to draw all three equations. You will see three planes, and all three intersect at a single point, viz. (1, 2, 3).
I don't see that any more can be said, particularly since you "do maths just as a hobby" and hence your exposure to and understanding of the pre-requisite mathematical concepts is an unknown quantity.