Find the intersection of pi1, pi2, pi3, and describe geometrically in each case.

pi1: x+y-z=3

pi2: 2x-y+z=5

pi3: x-2y+2z=6

ok, so this is my work (how does it look?)

1 1 -1 3

2 -1 1 5

1 -2 2 6

1 1 -1 3

2 -1 1 5

2 -4 4 12 << R3 x 2

1 1 -1 3

0 3 -3 -7 << R2 - R3

2 -4 4 12

1 1 -1 3 << R1 -R3

0 3 -3 -7

1 -2 2 6 << x (1/2)

0 3 -3 -3 << R1 - R2

0 3 -3 -7

1 -2 2 6

0 0 0 4

0 3 -3 -7

1 -2 2 6

final matrix

1 -2 2 6

0 3 -3 -7

0 0 0 4

so from above, would it be correct to say that there are no intersections? since 0z=4

but i'm confused because i don't think that it makes sense, since all three planes are not parallel.. meaning that there should be a point of intersection.. what did i do wrong here?