Inconsistent and Dependent SystemsDate: 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/ |
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