Thank you again for your explanation. I'm just about fully understanding what he was trying to teach us.

He first wrote the general form of a line on the board, Ax + By - C = 0

Then he wrote "two points determine a line".

Then.... (x1, y1) = (7,5) , (x2, y2) = (4,3) .....find A,B,C

he set it up two equations:

$\displaystyle 7A + 5B - C = 0$

$\displaystyle 4A + 3B - C = 0$

then he put these equations in matrix form.

After putting it in RREF form, we were left with:

A = -2C , B = 3C, and C cannot equal zero ...you showed me earlier that I can write this as a vector with one independent variable.

However, what does this vector represent? Is it supposed to represent a place where two lines intersect? Since I was given two points, it only seems as if I have one line, so how could it represent an intersection?

Furthermore, seeing an equation set-up as 7A + 5B - C = 0 seems so strange to me. Most of the time, the "A" and "B" are given, but in this case the x and y are given. Very, very hard for me to understand conceptually.

Ahhhh.