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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?

Respuesta :

Yep.....the slopes are the same

 

To see this......suppose 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

 

So....the slopes are exactly the same.......

 

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