Consistent Dependent, Consistent Independent, Inconsistent Linear Equations

Date: 03/27/2003 at 22:40:54
From: Rachael
Subject: Consistent and independent linear equations

I cannot find the definitions or differences between consistent and 
independent, consistent and dependent, and inconsistent linear 
equations. 

An example is:
State whether each system is consistent and dependent, inconsistent, 
or consistent and independent.

   3x+4y = 5
   2x-5y = 8

Date: 03/28/2003 at 00:43:45
From: Doctor Wallace
Subject: Re: Consistent and independent linear equations

Hello Rachael,

You can decipher the meanings by looking at what the terms mean in 
English. The words go in pairs, and each means the opposite of the 
other. They are used to describe the SOLUTION of a system.

The first pair is "consistent" versus "inconsistent."

Now, keep in mind that you are applying these to a system of linear 
equations. We say that a point is a "solution" to the system when it 
makes BOTH equations true, right? This is to say that there exists a 
point (or set of points) that "work" in one equation and also "work" 
in the other one. So we say that this point is CONSISTENT from one 
equation to the next.  

On the other hand, if there are NO points that work in both, then we 
say that the equations are INCONSISTENT. NO numbers that work in one 
are consistent with the other.

To sum up, a consistent system has at least one solution. An
inconsistent system has NO solution at all.

Now for the other pair. "Dependent" versus "Independent."

When a system is "dependent," it means that ALL points that work in 
one of them ALSO work in the other one. Graphically, this means that 
one line is lying entirely on top of the other one, so that if you 
graphed both, you would really see only one line on the graph, since 
they are imposed on top of each other. One of them totally DEPENDS 
on the other one.

When a system is "independent," it means that they are not lying on 
top of each other. There is EXACTLY ONE solution, and it is the point 
of intersection of the two lines. It's as if that one point is 
"independent" of the others.

To sum up, a dependent system has INFINITELY MANY solutions.
An independent system has EXACTLY ONE solution.

Now we can combine the terms.  Since we have 2 terms and 2 terms, you 
would think that there would be 4 possibilities:

    1. Consistent Dependent
    2. Consistent Independent
    3. Inconsistent Dependent
    4. Inconsistent Independent

However, 3 and 4 are not possible, since "Inconsistent" means no 
solution. Independent and Dependent BOTH mean there is a solution, so 
they can't ever go with Inconsistent because that would be 
contradictory.  

So really there are only three possibilities: Consistent Dependent, 
Consistent Independent, and Inconsistent.

We ordinarily don't even use "consistent" with dependent or 
independent, since once you know what these latter two words mean, you 
already know they are consistent, so it is enough to say the system 
is "dependent" or "independent."

We usually use the word "consistent" when we are more interested in 
indicating that the system does HAVE a solution, rather than 
indicating how many solutions it has.

I hope this clears it up for you. Don't hesitate to write again if you 
need further help with this or another question.

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