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What We're Doing When We Do Algebra

Date: 03/25/2002 at 03:11:54
From: Mike
Subject: Geometry

I do not understand how you graph x + y = 5 on a graph sheet.  

Date: 03/25/2002 at 09:32:49
From: Doctor Ian
Subject: Re: Geometry

You can find an introduction to graphing linear equations here:

   Graphing Linear Equations   

- Doctor Ian, The Math Forum   

Date: 03/25/2002 at 22:43:08
From: Mike
Subject: Geometry

I looked at the web site and think I have it. I believe it is as 

Y = 6x+1
Y = 6(0) + 5 = 5 (0,5)

Y = 2(2) + 1 = 5 (2,5)
Y = 3(-2) + 1 = 5 (-2,5)

Date: 03/26/2002 at 07:57:56
From: Doctor Ian
Subject: Re: Geometry

I'm having a little trouble following what you're doing here. Are you 
generating (x,y) pairs that satisfy the equation x+y = 5? If so, you 
need to check each pair as you test it:

  (x=0,y=5)  =>   x + y = 0 + 5 = 5          (yes)

  (x=2,y=5)  =>           2 + 5 = 7          (no)

The idea behind an equation like x+y = 5 is that the values of x and y 
are mutually constrained. So if you choose one value, the other value 
is determined. For example, if we _decide_ that x should be 2, then

      x + y = 5

      2 + y = 5        

  2 + y - 2 = 5 - 2

          y = 3        

That is, IF we set x = 2, then it MUST be true that y = 3. On the 
other hand, if we _decide_ that y should be 4, then 

      x + y = 5

      x + 4 = 5        

  x + 4 - 4 = 5 - 4

          x = 1

That is, IF we set y = 4, then it MUST be true that x = 1.     

So we've found two points that must be on the line: (2,3) and (1,4).  

If you were to generate more points, and plot them individually, you'd 
find that they all lie on a straight line:

       |  *
       |     *
       |       *
       |          *

What this means is that, having found any two of the points, we can 
use a ruler to fill in the rest. 

- Doctor Ian, The Math Forum   

Date: 03/27/2002 at 22:36:52
From: Mike
Subject: Geometry

I was just figuring the problem wrong. But I have the following 

  X+Y = 5
  2+Y = 5
  2+Y-2 = 5-2
  Y = 3

What if it is subtraction? Does it work the same?

  X-Y = 5
  7-Y = 5
  7-Y = 5+Y
  Y = 2

Would that be correct?  I don't have a particular problem that I'm 
working on like this, but just wanted to know how it would work if it 
was subtraction instead of adding.

Thanks for your help.

Date: 03/28/2002 at 08:51:46
From: Doctor Ian
Subject: Re: Geometry

Hi Mike, 

You came up with the right answer, which you can check by substituting 
both values into the original equation:

  x - y = 5

  7 - 2 = 5          (True)

although I'm not really able to follow your steps, perhaps because you 
combined some steps. I think you meant to do this:

       x - y = 5

       7 - y = 5                  Choose x = 7

   7 - y + y = 5 + y              Add y to both sides

           7 = 5 + y              Simplify

       7 - 5 = 5 + y - 5          Subtract 5 from each side

           2 = y

When you're first getting the hang of manipulating equations, it's a 
good idea to avoid trying to combine steps. For one thing, it's easy 
to slip up - in your case, you added a Y to one side, but forgot to 
remove it from the other. You seem to have cleared that up later on, 
but someone reading your steps wouldn't know that. A teacher would 
just see that you had written 

  7 - y = 5 + y

which is true only when y is 1, not 2.  

A very smart teacher of mine once told me that in algebra, there are 
really only two rules that you have to follow. The first is that you 
can't divide by zero. The second is that you can write down anything 
you want, so long as it's true. 

You ended up with the right value for y, but along the way you wrote 
down things that weren't true. As I said, this may not be a problem 
for you; but it would be too much to expect someone else who is trying 
to follow your work to assume that you know what you're doing.  

(Imagine that you're a teller in a bank, and at some point during the 
day you lend $50 from your register to another teller. Now, you know 
what you did; and the other teller knows what you did; and probably 
you'll get it all straightened out at the end of the day. But if an 
auditor comes by and wants to make sure that you've got the right 
amount of money, you're going to come up $50 short. And it won't help 
to say that you loaned the money to another teller, because the 
auditor can only go by what you've written down. If you want to lend 
$50 to another teller, you've got to have some kind of piece of paper 
that says that you did it, and that the other teller owes you $50.)

So this is algebra in a nutshell: We start out with an equation that 
we're told is true for some value of x, or some values of x and y, or 
whatever. But we would like to see it written in a friendlier way. So 
we apply transformations to the equation, to change its APPEARANCE 
without changing its MEANING.  

In English, we do this kind of thing all the time without even 
thinking about it. I might tell you: "Pat gave Chris a ride." Later, 
you might repeat it to someone else by saying: "Pat gave a ride to 
Chris." And the person you repeated it to might repeat it to someone 
else by saying: "Chris was given a ride by Pat."

Now, all of these mean exactly the same thing, even though they're 
expressed in different forms. Why choose one form over another? In 
some cases, it might be because of the way a question was asked:

  Who gave Chris a ride?
  Pat gave Chris a ride.

  To whom did Pat give a ride?
  Pat gave a ride to Chris.

  How did Chris get here?
  Chris was given a ride by Pat. 

In the same way, in response to the question, "What is y when x is 2?" 
we would like to change x + y = 5 into x = 5 - y. But in response to 
the question, "What is x when y is 2?" we would like to change it into 
y = 5 - x.  

All these equations mean exactly the same thing, even though the 
elements are arranged differently, just as with the sentences. 

Does this make sense? 

- Doctor Ian, The Math Forum   
Associated Topics:
High School Basic Algebra
High School Equations, Graphs, Translations

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