Determining Slope

Date: 05/19/99 at 01:47:17
From: Teresa Jaramillo
Subject: Determining the slope

(3,4) (4,6)
(18,-4) (6,-10)
(14,3) (-11,3)
(3/4,1) 3/4,-1)

Finding r  (9,r) (6,3) m = -1/3
(r,7) 11,r) m = -1/5

I don't understand how to do this. Please help me.

Date: 05/19/99 at 15:37:46
From: Doctor Rick
Subject: Re: Determining the slope

Hi, Teresa.

You're completely lost. Let's try to help you find your way. I'll draw 
you a map, I mean, graph.

Here is the first problem:

  y
  |
6 +               X (4,6)
  |              /|
5 +             / |2
  |            /  |
4 +     (3,4) X---+
  |           : 1 :
3 +           :   :
  |           :   :
2 +           :   :
  |           :   :
1 +           :   :
  |           :   :
0 +---+---+---+---+---+--> x
  0   1   2   3   4   5

The slope of the line joining (3,4) and (4,6) is the difference in y
coordinates divided by the difference in x coordinates, or "rise" 
(step height) over "run" (step width). The rise is 2 and the run is 1. 
The slope is therefore 2/1, or 2.

I'll do the next one without a graph. What is the slope of the line 
through (18,-4) and (6,-10)? The rise (difference in y) is

  (-10) - (-4) = -10 + 4 = -6

The run (difference in x) is

  6 - 18 = -12

The slope is

  rise/run = (-6)/(-12) = 1/2

We could just as well have switched the two points; the slope would 
come out the same: (6,-10) and (18,-4).

  rise = (-4) - (-10) = -4 + 10 = 6
  run = 18 - 6 = 12
  slope = rise/run = 6/12 = 1/2

Now let's do one of the last two problems. What is r such that the 
slope of the line through (9,r) and (6,3) is m = -1/3? I'll do it 
graphically first so you can get the idea, then I'll do it 
numerically.

  y           \
  |                 \
3 +                 (6,3) X           |
  |                     -1|     \     |
2 +                       +-----------O (9,r)
  |                       :     3     |     \
1 +                       :           |
  |                       :           |
0 +---+---+---+---+---+---+---+---+---+---+--> x
  0   1   2   3   4   5   6   7   8   9  10

The point (6,3) is marked X. I then drew a line with slope -1/3 
through this point: the rise is -1 (it falls instead of rising) and 
the run is 3. The other point (9,r) has x-coordinate 9, so it is on 
the vertical line through x = 9. Where the two lines intersect, at O, 
is the point we're looking for; its y-coordinate is r. Can you see 
what r is?

Now I'll do it numerically so we can be sure we have the exact answer. 
Let's switch the points: (6,3) and (9,r). The rise is r - 3 and the 
run is 9 - 6 = 3. Therefore the slope is (r-3)/3. If the slope is 
-1/3, we have the equation

  (r-3)/3 = -1/3

What is r? I will leave this to you to solve. I hope you get the idea 
now - if it's still confusing, we can try some more.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/