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What is a Line?

Check out this example:

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

The graph does LOOK good...

a graph of the line x + y = 3 and the point ( 2, 1 ) which LOOKS like it might be ON the line

But, it would be nice to know for sure!

So, we use some... Algebra!

It's easy.  Just take the line...

Let's plug (substitute) the point ( 2, 1 ) into the line x + y = 3

x + y = 3 which gives us 2 + 1 = 3 which gives us 3 = 3

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

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

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

Since we don't know how to graph it, we have to stick with the algebra:

Plug the point ( -1, 2 ) into the line 2x - 3y = 6

2x - 3y = 6 which gives us 2(-1) - 3(2) = 6 which gives us -2 - 6 = 6 which gives us -8 = 6 ?

No way, Dude!

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