What is the significance of adding a constant in augmented coefficient vectors?

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Square1
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What exactly does throwing the constant into your normal tuplet represent vs. a normal vector without the constant? (coefficient vectors = normal vector right?)

My notes don't provide an explanation for this. I can visualize what parallel normal vectors look like, but then they talk about parallel or not parallel augmented coefficient vectors which is where I get lost.

Thanks.
 
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a simple example:

the vectors (2,1) and (4,2) are parallel. suppose they actually represent coefficients in a set of linear equations:

2x + y = 0
4x + 2y = 0

the augmented vectors (2,1,0), (4,2,0) are also parallel. this means they represent "the same equation" (just scaled, in this case, by a factor of 2).

if we have:

2x + y = 0
4x + 2y = 1

the augmented vectors (2,1,0) and (4,2,1) are no longer parallel, which means the same pairs of (x,y) no longer satisfy both (in this case, there is no such pair at all).

adding an extra coordinate increases the dimension of "the space we're in" by 1. in 2 dimensions (the plane), if two lines are not parallel, they have to intersect. in 3 dimensions, two lines can be "not parallel" and still not intersect.

if two augmented vectors are not parallel, it means they represent two different equations (two different constraints on the solutions space). if two augmented vectors are parallel, one of them is redundant (this is the notion we seek to capture with the idea of linear independence, when we are considering two or more equations).

this is the "row-based" way of looking at things (focused on the solutions). the "column-based" way of looking at things focuses on the "images" (what happens to the elements of the solution space). in the equation:

Ax = b

the columns (of A) determine which b's we can get, the rows (of A or A|b) determine which x's we can use.