Question on homogeneous linear systems

[mpm]

I have a homework question that I don't really understand what they are asking.

The book I am using is terrible so I was hoping someone could shed some light.

Question:

Give a geometric explanation of why a homogeneous linear system consisting of two equations in three unknowns must have infinitely many solutions. What are the possible numbers of solutions for a nonhomogeneous 2 x 3 linear system? Give a geometric explanation of your answer.

If anyone could help me on this I would appreciate it.

mpm

 

[asdf1]

2 equations in 3 unknown must have infinitely many solutions, because the only way you could have one singular solution is to have 3 equations for each unknown...
i think that there's some matrix(or is it determinant?) formula (i forgot what it's called, because it's been a long time since I've touched that...) that you can you use to prove...

 

[asdf1]

or you could draw pictures of the planes in space, and find out that there are infinitely many solutions...

 

[amcavoy]

Well let's say you reduce it to the form:

$$\left[\begin{array}{ccc|c}1 & 0 & a & 0 \\ 0 & 1 & b & 0\end{array}\right]$$

The general solution is then:

$$x_3\begin{bmatrix}-a \\ -b \\ 1\end{bmatrix}$$

...which has a solution for each value of $x_3$.

 

[mpm]

Ok I think I understand the homogeneous part of the question. Can anyone help on the nonhomogeneous part?