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Hidden in x-y Plane Sight

Date: 11/29/2011 at 23:28:19
From: Renee
Subject: x=4

Why is this a vertical line?

   x = 4

I don't understand.



Date: 11/30/2011 at 10:31:11
From: Doctor Ian
Subject: Re: x=4

Hi Renee,

Think about a line that isn't horizontal or vertical, like 

   2x + 3y = 8

A "solution" to any equation is an assignment of values to variables that
makes the equation true. In this case, these values make the equation true:

   2(1) + 3(2) = 8

So (x = 1, y = 2) is a solution to the equation. More simply, we might
write just (1,2). Does this make sense so far?

Now, for this line, every time we change x, we have to change y, too. For
example, if we increase x to 2, we get

   2(2) + 3y = 8

          3y = 4

           y = 4/3

So (2,4/3) is also a solution.

But now think about this line:

   0x + 2y = 8

Since the coefficient of x is zero, it doesn't matter what x is. As long
as y is 4, we can choose any x, and it will be a solution. So (1,4),
(5,4), (-6,4), and so on, are all solutions. 

If you graph those, you get a horizontal line. Make sense?

Now, what if we have this?

   2x + 0y = 6

In this case, as long as x is 3, we can choose any value for y. So (3,1),
(3,-4), and (3,19.8) are all solutions.

If you graph those, you'll get a vertical line. Make sense?

Now, in the horizontal case, we can simplify the equation by noting that
0x is always 0, to get

   2y = 8

So this kind of equation gives a horizontal line: you can't see the x
term, 0x -- but it's implied.

In the vertical case, we can simplify the equation by noting that 0y is
always 0, to get

   2x = 6

So this kind of equation gives a vertical line: you can't see the y 
term -- 0y, but it's implied.

Does this all make more sense now?

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
Middle School Graphing Equations

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