Let's take the lines passing through the sets of points (x1, y1) and (x2, y2) and (x'1, y'1) and (x'2, y'2)
The slope of the first line passing through points (x1, y1) and (x2, y2) is given by s = (y2 - y1)/(x2 - x1)
The slope of the line passing through the points (x'1, y'1) and (x'2, y'2) is given by s' = (y'2 - y'1)/(x'2 - x'1)
If the two lines are parallel we have s = s'.
We'll take another example in which we'll prove that 2 lines given by their points, are not parallel.
We'll take the line through (-1,1) and (5,2) and we'll verify if it is parallel to the line through (-1,4) and (2,3).
We know that 2 lines are parallel if their slopes are equal.
Since we know the points through the lines are passing, we'll calculate the slopes.
Slope of line through the points (-1,1) and (5,2) is:
m1 = (2-1)/(5+1)
m1 = 1/6
Now, we'll consider the next two points (-1,4) and (2,3).
m2 = (3-4)/(2+1)
m2 = 1/3
m1 is different from m2.
Therefore, the given lines are not parallel.