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### Inconsistent and Dependent Systems

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Date: 12/07/97 at 22:03:12
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.
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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/
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Associated Topics:
High School Linear Equations

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