Q:

Augmented Matrices of Consistent Linear Systems

So my textbook for Linear Equations has problems referencing augmented matrices, but I can't find where it talks about it. I did find a few examples but I want to know what an augmented matrix is and why the following are or are not. Wikipedia was not helpful... (also, this is my first day of class and we only went over the syllabus, but I want to get some stuff under my belt so I can ask my TA's questions in Wednesday discussion). 
\begin{bmatrix}1 & h & 2\\-5 & 20 & -12\end{bmatrix}
The above matrix is the augmented matrix of a consistent linear system if $h\ne4$
\begin{bmatrix}1 & 4 & -2\\2 & h & -4\end{bmatrix}
The above matrix is the augmented matrix of a consistent linear system.
\begin{bmatrix}-8 & 24 & h\\2 & -6 & 7\end{bmatrix}
The above matrix is the augmented matrix of a consistent linear system if $h\ne-28$
Then the problems from the book I don't have the answers too, but this one builds on augmented matrices:
\begin{bmatrix}1 & -3 & 7 & h\\0 & 2 & -8 & g\\ -2 & 4 & -6 & k\end{bmatrix}
Like what? How do I even start? Can I safely assume that x-3y+7z=g?
Edit What is a "consistent linear system"?

A:

An augmented matrix contains the coefficients of a system of linear equations.  It consists of the coefficients of the LHS (all terms with variables), with the coefficients of the RHS (all terms without variables) "stuck on" at the right - hence the name "augmented".  It is common to draw a vertical line separating the LHS from the RHS.  So for
$$\eqalign{2x+3y=4\cr 5x+6y=7\cr}$$
I would write the augmented matrix as
$$\left(\matrix{2&3\cr5&6\cr}\ \bigg|\ \matrix{4\cr7\cr}\right)\ .$$
Some people don't draw the vertical line - from your examples it looks like this is the case with your instructor - but I find this confusing.  In any case you can always imagine the line to be there :)

