Are line L, through (6,1) and (9,7) and L2 through (-9,8) and (10,9) parallel
or do they intersect?

First let's draw those lines and see how they look:



They certainly look like they intersect and are not parallel.  But "looking and
seeing", though it is a good check, it is not considered good mathematics.  So
we calculate the slope of each line.

Unless their slopes are the same, they aren't parallel and will intersect.

Use the slope formula for the green line, where
(x1,y1) = (6,1) and (x2,y2) = (9,7)

     y2 - y1     7 - 1     6
m = --------- = ------- = --- = 2
     x2 - x1     9 - 6     3

Use the slope formula for the blue line, where
(x1,y1) = (-9,8) and (x2,y2) = (10,9)

     y2 - y1      9 - 8          1        1
m = --------- = ---------- = -------- = ----
     x2 - x1    10 - (-9)     10 + 9     19 

Well, 2 and 1/19 are certainly not equal so
that proves they are not parallel and therefore
they intersect.

Incidentally they intersect at the point (, )

Edwin