Hi

More simple

R1+R2 => 3x=8

2R1+R3 => 3x=12

There is effectively no intersection

3 planes do not always have an intersection point

Think about 2 planes : or they are parallel or they are secant along a line

In this case the intersection with a 3rd plane is the intersection between the line and this 3rd plane : you can see that it can be empty if the line is parallel to the plane

The intersection between pi1 and pi2 is a line that can be defined by

x=0t+8/3

y=t

z=t-1/3

Its direction is given by u(0,1,1)

A normal vector to pi3 is given by v(1,-2,2)

The dot product between u and v is 0, which means that the line is parallel to pi3