Inconsistent and Dependent Systems

Date: 12/07/97 at 22:03:12
From: Adam Shoup
Subject: Systems of equations

Write a second linear equation for y = 2x + 4 that would create an 
inconsistent system. Also write a second linear equation that would 
create a dependent system.

I have definitions of both, but I don't know how to do the problem.

Date: 12/08/97 at 09:28:19
From: Doctor Pete
Subject: Re: Systems of equations

Hi,

So, you've got the definitions, but do you know what they mean?  
An inconsistent (or overdetermined) system is one without a solution.  
A dependent (or underdetermined) system is one where there is more 
than one solution.  Let's give some simple examples:

Inconsistent/overdetermined:

     (I1)  0 = 1

     (I2)  y = 1  and  y = 2

     (I3)  x = y  and  x = y + 1.

Dependent/underdetermined:

     (D1)  x = y

     (D2)  y = 3x  and  2y = 6x

     (D3)  x + 2y = 0.

Now, some comments on each of these. (I1) is a false statement, and 
even though it is not strictly a system, or even a statement about 
unknown quantities (like x or y), it is inconsistent because it is 
false! In (I2), it is a system, but it cannot be true, because it 
implies 1 = 2. In (I3), we again have a similar situation; x = y = y+1 
implies 0 = 1. 

In (D1), there are infinitely many solutions, like x = y = 5, or 
x = y = -1. In (D2), the second equation is the same as the first when 
divided by 2, so it provides no additional information about x or y.  
Finally, (D3) is the same as saying x = -2y, so it too is dependent 
for the same reasons (D1) is.

Now, say you are given y = 3x + 5.  You want to write another equation 
which, with the given equation, makes an inconsistent system. There 
are lots of ways to do this; one way is to say

     y = 3x + 6,

so then we are forced to conclude that 3x + 5 = 3x + 6, or 5 = 6.  
Even easier,

     y = 3x

is also another possible way to do this. Now, suppose you want to make 
a dependent system. Well, you can do it by following example (D2), and 
multiply both sides by some number.  2 sounds good:

     2y = 6x + 10.

Wow, that was pretty easy. Hmm... You could also do it another way, by 
adding or subtracting some number from both sides:

     2y - 4 = 6x + 6.

-Doctor Pete,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/