Ben and phobeo are finding the slope of a line. ben chose two points on the line and used them to find the slope. phoebe used two different points to find the slope. did they got the same?

Yepthe slopes are the same

To see thissuppose that  the slope  of the line is  p/q

And let's suppose that Ben uses the points (a, b) and (c, d)so the slope between his points is

      [ d - b]   / [c - a]   = p/q

Which  implies that   [d - b]  = p     and  [c - a] = q      (1)

Then the points that Phoebe uses can be described by (a+mq, b +mp)  and  (c + nq,  d + np)   where  m and n are just some multipliers

Then the slope of Phoebe's line is 

  [ (d + np)  - ( b + mp)] / [ (c +nq )  - ( a + mq)]    =

[p(n - m) + ( d - b) ] / [ q(n - m) + (c - a)] 

Now, let (n - m) = r   and we have

[(d - b) + pr] / [ (c - a) + qr]    and substituting from (1), we have

[p + pr] / [ q + qr ]  =

[p(1 + r) ] /  [q (1 + r) ]  =

p/q

Sothe slopes are exactly the same