The Idea Behind Simultaneous Equations

Date: 10/14/2001 at 14:52:30
From: Laura-Lee
Subject: Systems of equations

I'm having a hard time learning the linear combinations method and the 
substitution method in algebra II. Is there some kind of trick to 
memorizing them?

Date: 10/15/2001 at 12:22:37
From: Doctor Ian
Subject: Re: Systems of equations

Hi Laura-Lee,

Actually, trying to memorize them would be about the worst thing you 
could do. The problem with memorizing things is that if you forget 
something that you've memorized, you leave yourself with no options 
except guessing what it's supposed to be.  

You'd be much better off trying to _understand_ how the various 
solution methods work. Better still, you can try to get a solid 
understanding of just what is going on when you have two simultaneous 
linear equations. Once you have that, you can choose from any of a 
number of possible solution methods, or even make up your own on the 
fly. 

When you have an equation like

  3x + 2y = 5

there are infinitely many pairs of values for x and y that can make
the equation true.  For example,

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

so (x = 1, y = 1) is a solution to the equation.  Also, 

  3(1/3) + 2(2) = 5

so ( x = 1/3, y = 2) is also a solution.  

An equation like this is a little like the volume control on a stereo.  
As you move the control to different positions (change the value of 
x), you increase or decrease the volume of the music (increase or 
decrease the value of y). 

Sometimes you can make the music as soft or as loud as you want. But
there are other times when you have to meet some constraints. For
example, perhaps you want to be able to hear the music in the next
room (so it can't be too soft), but you don't want your mom to 
complain about it (so it can't be too loud).

This is sort of like what happens when you have more than one 
equation, and all of them have to be true at the same time. For
example, 

  3x + 2y = 5

and

   5x + y = 13

As we've seen, if we consider either one of these equations by itself,
it has an infinite number of possible solutions. But what if we
consider them together? Then it turns out that there is only way to
make _both_ equations true with a single pair of values, and that is
to set x = 3 and y = -2. 

This can seem very mysterious, until you think about what's really
happening here. Each of these equations describes a line. Each point
on a line corresponds to a solution of the equation of the line. And
if two lines aren't parallel, then they intersect at exactly one
point. This point is the solution of the simultaneous equations.

(If I tell you that family A is driving on Interstate X, and family B 
is driving on Interstate Y, they could be just about anywhere. But if 
I tell you that their cars collide, that tells you exactly where they 
were at the time of the collision!)

When you think about it, this is a little like what you do with your 
eyes when you judge the distance to an object. An object that takes up 
a certain amount of space on your retina might be very small but very
close; or very large but very far away; or somewhere in between.  

With one eye, you can't really tell which of these is the case,
because there are infinitely many situations that could produce the
same image on one retina:

                       large
         small         and
         and           far?
         close?          .
                  .      |
           .      |      |
    @      |      |      |         One eye can't distinguish
           .      |      |         among these possibilities...
                  .      |
                         .
           or somewhere
           in between?

That is, there is an 'equation' that relates a given size to a given
distance, and it has infinitely many 'solutions'. 

But as soon as you bring a second eye into the picture, there will be
only _one_ distance that 'solves' the 'equations' for both eyes at the
same time:

                eye 2: "It's somewhere on this line"
     @           .
       .       .
         .   .
           x    both eyes:  "It must be here!"
         .   .
       .       .
     @           .
               eye 1: "It's somewhere on this line"


That's the basic idea behind simultaneous equations. 

With that in mind, try reading about the combination and substitution 
methods again, and see if they make more sense to you.  If they don't, 
write back and I'll try to find another way to help you understand why 
those methods work.

I hope this helps. 

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