As I understand it, there are three possibilities for linear systems; no solution, unique solution, or infinitely many solutions.
(i) For unique solution, there is only one intersection, a point where the given lines meet/intersect.
(ii) For infinitely many solutions, we get parameters, meaning any value on the said parameters works just fine because the lines occupy same space/line.
(iii) For unique solution, we don't have any intersection, the lines are parallel, no more no less, none of the lines are little off or whatnot, they are exactly parallel meaning they never ever meet.
Now, what got me confused is, when I was reading about a linear system with no solutions I noticed they used the following words to describe the result: "Parallel and distinct". Why would they use the word "distinct", because if e.g. two lines are parallel then they are distinct anyway? (or am I missing something, and if so are there cases where two parallel lines aren't distinct?) The said problem is x+y=4 and 0=-6, this troubles me because this is the only instance where they've used the word "distinct" while on other problems they simply say "parallel"
Best Answer
They probably say parallel and distinct just to rule out the possibility of having two of the same line. If you have two copies of the same line, they are probably considered by the authors to be parallel, but are definitely not distinct.
If you have 0=-6 as part of the system, then you've come to a contradiction, which means that you started with two "parallel" distinct lines.