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(In)consistent and (In)dependent Systems

Date: 01/02/2007 at 15:16:39
From: Molly
Subject: (In)consistent and (in)dependent systems

I was wondering why when solving systems of equations the types of 
systems are referred to as "consistent/inconsistent" and

I understand which type of system each one defines, I'm just wondering
why those vocabulary words were chosen.  I'm not confused so much as I
am curious.

Date: 01/02/2007 at 23:37:42
From: Doctor Peterson
Subject: Re: (In)consistent and (in)dependent systems

Hi, Molly.

Part of the answer can be found here:

  Consistent Dependent, Consistent Independent, Inconsistent Linear

I'll give you my explanation of the reason for the specific terms,
which I think works pretty well (and makes a fun story to tell!).

First, imagine you're a judge, and two witnesses to a traffic accident
come before you.  If they say

  1. A red-haired lady ran the red light and hit the pedestrian.
  2. A bald man ran the red light and hit the pedestrian.

what would you say?  The testimonies are INCONSISTENT--they can't both 
be true!  We can't use them to convict anyone.

That's what happens when a system of equations is inconsistent--they 
can't all be true (at the same time), so there is no solution.

If the system (or witnesses) are CONSISTENT, that means they DO fit
together, and there IS at least one solution.

Now suppose they say this:

  1. A red-haired lady ran the red light and hit the pedestrian.
  2. A red-haired lady ran the red light and hit the pedestrian.

The second seems to have copied the first; he adds no information.  We
can say his testimony is DEPENDENT on the other; we don't have two
independent witnesses.  As a result, we don't have enough evidence for
a conviction.

When a system of equations is dependent, it means that we can 
construct one of them from the others (by adding and multiplying, as
you do when solving by elimination), so that equation doesn't add
anything to our knowledge of the problem.  If we have three equations
in three unknowns, but they are dependent, we don't really have three
"independent witnesses", so there is not enough information to convict
--that is, to find a single solution.  Any red-haired lady might have
done it.

What the judge likes to see is a pair of witnesses who present 
different sides of the story, and add up to a complete identification
of the accused:

  1. A red-haired lady ran the red light and hit the pedestrian.
  2. A six-foot-tall woman with alcohol on her breath knocked the
     man over.

Now we have two INDEPENDENT witnesses whose testimonies are CONSISTENT
--we can put them together and get the information we need.

When a system of n equations in n unknowns is both consistent (not
contradictory) and independent (not duplicating information) then we
will have exactly one solution.  Each equation is a single witness,
giving one "line" of evidence; with two equations in a plane, the two
lines taken together "point" to a single solution.  Similarly, in
three-dimensional space, three equations represent three planes, and
when they are consistent and independent, they will meet in only one
place.  Guilty!

Does that make it clearer?

- Doctor Peterson, The Math Forum 

Date: 01/03/2007 at 09:06:49
From: Molly
Subject: Thank you ((In)consistent and (in)dependent systems)

Wow, thanks Dr. Peterson.  That makes complete sense and now I 
actually understand the words (and thus the systems) instead of just
memorizing them!!!
Associated Topics:
High School Definitions
High School Linear Equations

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