Type of systems of gauss-jordan elimination product

[Random-Hero-]

Homework Statement

Using Gauss-Jordan elimination determine the intersection of the following systems. Identify the type of system geometrically.

a) 3x-2y+z=0
4x-5y+7z=0
6x+3y=0

b) x-2y+3z=0
x+y-z=4
2x-4y+6z=5

c) x+y+2z=-2
3x-y+4z=6
x+2y=-5

The Attempt at a Solution

I got the following intersections for the above systems.

a) (0,0,0)
b) (0,0,1)
c) (1,-3,0)

However, how do I "identify the system geometrically"? Also, how would identify the type of system and verify my solution using the normals of the planes?

I'm having a lot of trouble with this, if anyone could help me out I'd really appreciate it!
Thanks in advance!

 

[Mark44]

Homework Statement

Using Gauss-Jordan elimination determine the intersection of the following systems. Identify the type of system geometrically.

a) 3x-2y+z=0
4x-5y+7z=0
6x+3y=0

b) x-2y+3z=0
x+y-z=4
2x-4y+6z=5

c) x+y+2z=-2
3x-y+4z=6
x+2y=-5

The Attempt at a Solution

I got the following intersections for the above systems.

a) (0,0,0)
b) (0,0,1)
c) (1,-3,0)

However, how do I "identify the system geometrically"? Also, how would identify the type of system and verify my solution using the normals of the planes?

I'm having a lot of trouble with this, if anyone could help me out I'd really appreciate it!
Thanks in advance!

You should check your work on b. (0, 0, 1) is not a point on any of the three planes. Your answers for parts a and c seem to be correct, but it's possible that there are more solutions than those you show.

As for identifying the type of system, each system of equation you show represents three planes in space. The planes can intersect a) not at all (no single point of intersection common to all three planes), b) in a single point, c) in a line (two planes are identical but not parallel to a third), and d) in a plane (all three planes are identical).

As for verifying your solution using normals, I believe this is meant to be used when you have concluded that the planes are parallel. If they are parallel, their normals will be multiples of one another.