Question

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?

Answer 1

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.......