In geometry, we say that a line is DEFINED by two points:

  But, a line isn't just made up of those two points.  It's made up of a TON
  of points -- an infinite number, in fact!

  Check out these lines:

  The equations of these line critters usually have an x guy and a y guy...

  Here are some examples:

  In the next sections, I'll be telling you more about these equations and
  you'll be learning about graphing and much, much more!  Tell them
  what they'll be learning, Bob!

  Welllll... In the next exciting and action packed lessons, our lucky
  students will be learning about

Equations of lines
Slopes
Graphing
and
Finding equations

  Information like this would usually sell for $500...  But, for our students
  today we have a special deal.  All of these lessons and more for the just
  14 easy payments of $59.99!

  (I really need to stop watching infomercials while I'm writing these
  lessons.)

  So, how can we be sure that a specific point is really ON a line?

  Graphing could be one way to get a general idea... But, there are two
  problems with this

  Check it out:

Is the point ( -1, 2 ) on the line 2x - 3y = 6 ?

  Here's the graph:

  But, what if it's not so easy to be sure?

  Check out this example:

Is the point ( 2, 1 ) on the line x + y = 3  ?

  The graph does LOOK good...

  But, it would be nice to know for sure!

  So, we use some... Algebra!

  It's easy.  Just take the line...

x + y = 3

  and plug the point in...

x + y = 3 with ( 2, 1 )

2 + 1 = 3 ?
3 = 3

The point works -- so, it's ON the line!

  Let's try the algebra with the first example we did:

  Plug the point in...

2x - 3y = 6 with ( -1, 2 )

2(-1) - 3(2) = 6 ?
-2 - 6 = 6 ?
- 8 = 6 ?
No way, Dude!

So, ( -1, 2 ) is NOT on the line 2x - 3y = 6.

  Check out one more example:

  We'll need to do the algebra:

x - 2y = -4 with ( -2.5, .7 )

-2.5 - 2(.7) = -4 ?
-2.5 - 1.4 = -4 ?
- 3.9 = -4 ?
Doh!  Close, but it just isn't happening!

So, ( -2.5, .7 ) is NOT on the line x - 2y = -4.

  The key is to ALWAYS use algebra to really be sure!

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