Why Does Adding Equations to Solve a System Work?

Date: 02/10/2007 at 16:23:55
From: Bill
Subject: adding equations

I don't understand why you can add two equations together.  For 
equations with x and y as variables they represent lines.  So are you 
adding the two lines together?  I understand HOW you can do this and 
eliminate one variable but I don't understand WHY you can do it at 
all.  Thanks for the explanation.

Equations are not numbers.  What does the resulting equation represent 
in terms of the two lines that the equations represent on a graph?  
-x + y = 10 added to x + y = 10 yields 2y = 20 or y = 10.  I don't see 
how this works.  I don't understand the concept involved.

Date: 02/10/2007 at 21:48:01
From: Doctor Ian
Subject: Re: adding equations

Hi Bill,

Good question!  I don't think this gets enough attention in most math
classes. 

First, let's think about what it means to add two equations.  Suppose
I have 

  a = b

and

  c = d

I can add the same thing to both sides of an equation, without
changing the truth of the equation, right?  I could, for example, do this:

  a + 1 = b + 1

So far, so good?  I could also do this:

  a + c = b + c

This works no matter what c is.  But suppose I'm in some context where
I know that 

  c = d

Since c and d are equal, I can use either one wherever I could use the
other, right?  (That's what it means for two things to be equal--
they're interchangeable.)  So I can write

  a + c = b + d

and I still have a true statement.  But what did I do?  Starting with

  [1]   a = b

  [2]   c = d

I added the left sides and the right sides to get 

  [3]   a + c = b + d

Do you see why this has to give me another true statement?  And
subtraction works for the same reason, since we can always multiply
the second statement by -1 on each side:

  [1]       a = b

  [2]       c = d

  [2']     -c = -d

  [3]   a + -c = b + -d

  [3']   a - c = b - d

So, that covers the MECHANICS of adding and subtracting equations.  It
just says we can add or subtract the right and left sides of 
equations, and under any conditions where the original equations were
true, the new one will also be true. 

But what does this MEAN, when we start with two equations?  Let's take
a look.  Here are two lines:

  y = 3x + 4 

  y = 2x + 5 

Suppose we plot these lines.  What would we see?  We would see two
lines that move together, intersect at some point, and then move apart
again, sort of like

     *                      @
           *             @
                 *    @
                   @   *
                @            *
             @                    *
          @
       @

So, suppose we drop some vertical line segments to connection the lines, 

                               @
     *                      @  |
     |     *             @  |  |
     |     |     *    @  |  |  |
     |     |     | @   *    |  |
     |     |    @            * |
     |     | @                    *
     |    @
     | @


What do these show us?  For any value of x, the height of the line
segment above tells us what we would get if we 

  1) Evaluated y = 3x + 4
 
  2) Evaluated y = 2x + 5

  3) Took the absolute value of the difference

What's interesting is that, at some point--the point where the lines
intersect--the difference will be zero.  How can we find that?  By
subtracting our two expressions for y:

  y - y = (3x + 4) - (2x + 5)

      0 = x - 1

And this tells us that the lines intersect at x = 1.  That is, by
subtracting the equations, we find the point of intersection. 

Does that make sense?  

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

Date: 03/06/2007 at 20:25:18
From: Bill
Subject: Thank you (adding equasions)

Thanks, that makes perfect sense.