Tables, Graphs, and Equations

Date: 09/04/2001 at 19:41:20
From: Chelsea
Subject: Geometry/Algebra Breakdown

Here is my problem:

Make a table of values to determine the coordinates of three points 
that lie on the graph of y = 3x+5. Use the headings x, 3x = 5,y, and 
(x,y).

A. Use these three points to draw the line in a coordinate plane.

B. Make a table of your values to determine three points on the graph 
   of y = -2x - 10. Then draw the line in the same coordinate plane.

C. Based on your graphs, what is the point of intersection of the 
   lines with equations y = 3x = 5 and y = -2x -10. 

I don't even understand the first sentence. :( 
Thank you so much.

Date: 09/05/2001 at 10:54:42
From: Doctor Ian
Subject: Re: Geometry/Algebra Breakdown

Hi Chelsea,

A table is just a structure that relates sets of items. For example, 
you could construct a table that shows the height and weight of each 
student in your class:

  Name     Height     Weight
  ------   --------   --------
  John     5'6"       135 lbs
  Sally    5'2"       115 lbs
  Tom      5'10"      170 lbs

The way you read a table like this is to look at the left side to find 
the person you're interested in (for example, Sally), and then read 
across to find the things you want to know about her:

  Name     Height     Weight
  ------   --------   --------
  John     5'6"       135 lbs
  Sally    5'2"       115 lbs <= Sally is 5'2" tall and weighs 115 lbs
  Tom      5'10"      170 lbs

In your assignment, the leftmost column will contain numbers instead 
of names, and the other columns will contain values obtained by 
putting those numbers into some expressions. 

This _looks_ complicated, but it's really the same sort of thing that 
we did to make our table of heights and weights. To make a new entry 
in the table, we take the next student - Veronica - and stand her next 
to some marks on the wall. We read the closest mark, and write it in 
the second column:

  Name     Height     Weight
  ------   --------   --------
  John     5'6"       135 lbs
  Sally    5'2"       115 lbs
  Tom      5'10"      170 lbs
  Veronica 5'4"

Then we put her on a scale, which is another set of marks, and again 
read the mark that is closest to the pointer on the scale:

  Name     Height     Weight
  ------   --------   --------
  John     5'6"       135 lbs
  Sally    5'2"       115 lbs
  Tom      5'10"      170 lbs
  Veronica 5'4"       108 lbs

In the case of numbers, instead of making 'measurements' we 'evaluate' 
expressions or functions. For example, instead of measuring the 
'height' of a number, we might multiply it by itself and see what we 
get:

  Number   Number times itself
  ------   -------------------
    1         1
    2         4
    3         9

We can write this more compactly as

  x        x * x
  ------   -------
  1        1
  2        4
  3        9
  
As before, we can have more than one 'measurement' corresponding to 
each number on the left:

  x        x * x    x + 2
  ------   -------  -------
  1        1        3
  2        4        4
  3        9        5

You've been told that your table has to look like this:

   x      3x+5   y      (x,y)
   -----  -----  -----  -----

And you've been told to make three entries in this table, which means 
you have to pick three values for the leftmost column. Sometimes there 
are good reasons for choosing particular values, but in this case, we 
can just choose the first values that jump into our heads:

   x      3x+5   y      (x,y)
   -----  -----  -----  -----
   0
   1
   2

Now we have to fill in the rest of the table. If the value of x is 0, 
what is the value of 3x+5?  3*0+5 = 5, so 

   x      3x+5   y      (x,y)
   -----  -----  -----  -----
   0      5
   1
   2

If the value of x is 1, what is the value of 3x+5?  3*1+5 = 8, so

   x      3x+5   y      (x,y)
   -----  -----  -----  -----
   0      5
   1      8
   2

I'll let you finish that column. The next column is easy, and a little 
pointless, since if we're graphing the line y=3x+5, we can just copy 
the values from the second column:

   x      3x+5   y      (x,y)
   -----  -----  -----  -----
   0      5      5
   1      8      8
   2

Note that a cleaner way to make the table would be

   x      y (=3x+5)  (x,y)
   -----  ---------  -----
   0      5      
   1      8      
   2

The final column is also easy, since we're just combining the values 
in the first and second columns:

   x      3x+5   y      (x,y)
   -----  -----  -----  -----
   0      5      5      (0,5)
   1      8      8      (1,8)
   2

The final step is to plot the values in the rightmost column on a 
graph. To plot a point like (2,3), you start at the origin of the 
graph (the place where the axes cross), move 2 units to the right, and 
then move 3 units up. Then you make a mark at that location:

         -
         -   x    (2 over, 3 up)
         -   
         -
   | | | + | | | | |
         -
         -
         -

If the first value is negative, you move to the left instead of to the 
right. If the second value is negative, you move down instead of up:

  (-2,3)  -  (2,3)
      x   -   x 
          -   
          -
    | | | + | | | | |
          -
          -
      x   -   x
 (-2,-3)  -  (2,-3)

If you plot the points from your table, you will find that they lie on 
a straight line, e.g.:

         -           x
         -       x
         -   x    
         -   
         -
   | | | + | | | | | | |
         -
         -
         -

(Note that this isn't _the_ line for your table, just an illustration 
of what it looks like when three points lie on a line.)

Then you do the whole thing over again for a different table:

   x      -2x-10  y      (x,y)
   -----  ------  -----  -----

You can pick the same values for x (0, 1, 2) or a different set of 
values.  

After you've constructed your second table and plotted your second set 
of points, you'll have a second line, e.g.:

         -           x
         -       x
   @     -   x    
         -   
         -
   | | | + | | | | | | |
       @ -
         -
         -
         -
         - @

If you use a ruler to 'fill in' the lines, you'll see that they 
intersect at some point (*).

         -           x
         -       x
   @     -   x    
         -   
     *   -
   | | | + | | | | | | |
       @ -
         -
         -
         -
         - @

You can then use the axes of the graph to identify the location of 
this point. In the graph above, it occurs at (-2,1).  In your graph it 
will probably occur somewhere else.  

Okay, so you're probably asking yourself: What is the _point_ of doing 
this? 

Well, if you took algebra last year, then you probably had to solve 
pairs of linear equations, e.g., 

  2x + 3y = 15
  6x - 4y =  6

You probably did it by substitution:

   1.  2x + 3y = 15

            2x = 15 - 3y

             x = (15 - 3y)/2

   2.              6x - 4y = 6

       6((15 - 3y)/2) - 4y = 6

           3(15 - 3y) - 4y = 6

              45 - 9y - 4y = 6

                   45 - 6 = 9y + 4y

                        39 = 13y
 
                         3 = y

   3.  x = (15 - 3y)/2

         = (15 - 3(3))/2

         = (15 - 9)/2

         = 6/2

         = 3

   4.  (x,y) = (3,3)

or by some other algebraic method, e.g., 

   1.  2x + 3y = 15

            3y = 15 - 2x

             y = (15 - 2x)/3

   2.    6x - 4y = 6

          6x - 6 = 4y

      (6x - 6)/4 = y

   3.           y = y  

      (15 - 2x)/3 = (6x - 6)/4

       4(15 - 2x) = 3(6x - 6)

          60 - 8x = 18x - 18

               78 = 26x

                x = 3

   4.     y = (15 - 2x)/3

            = (15 - 2(3))/3

            = (15 - 6)/3

            = 9/3

            = 3

   5. (x,y) = (3,3)

The point of this exercise is to show you another way to think about 
this kind of problem. You can use algebra to find a pair of values 
(x,y) that satisfy a pair of linear equations, as I just did above.  
You can _also_ graph the equations and find the point where the graphs 
intersect, which gives you a pair of values (x,y) that satisfy the 
equations.  

(Make sure you understand why this is the case. Each point on the 
first graph is a pair of values that satisfies one equation. Each 
point on the second graph is a pair of values that satisfies the other 
equation. If a point lies on both graphs, what does it mean? How many 
such points are there?)

These are just two ways of solving the same problem. Why learn two 
different methods? For the same reason that it's good to know more 
than one route home from school. If one route is blocked, you can use 
the other. 

Sometimes it's easier to use equations; sometimes it's easier to use 
graphs. If you know how to use both methods, you can pick the one that 
best suits the problem you're trying to solve. If you only know one 
method, you might find yourself in the position of having to drive 
through Chicago to get from New York to Boston, so to speak. 

Does this help?  Write back if you'd like to talk about this some 
more, or if you have any other questions. 

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