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Dear ladies and gentlemen,

over time I noticed I (and other) again and again have problems solving "systems of linear equations". It seems depending of the steps one chooses, we get different results!! How can that be? Should one not always get the same results no matter which path he goes down? What am I missing? Are there maybe rules I don't know and use even though I shouldn't?

I want to give you an example. The set of equations is from a state-price security calculation (see "State Preference Approach") which we shall solve using the Gaussian elimination:

I: 2P(1) + 2P(2) + 2P(3) =1,6

II: 3P(1) + 0P(2) + 1P(3) =1,0

III: 0P(1) + 2P(2) + 1P(3) =0,8

The solution shall be: P(1) = 0,2 P(2)=0,2 and P(3)=0,4

Not only got I different numbers on the first try that totally went in "into space", I want to write down for you my second approach to check for mistakes:

II-III = IV = 3P(1) - 2P(2) = 0,2 --> 2P(2) = 3P(1) - 0,2 (this far I'm with the solution) then I simply plug in the result in the following lines:

in III: 4P(1) = 1 --> P(1) = 0,25

in II: 0,75 + P(3) = 1,0 --> P(3) = 0,25

in I: 0,5 + 2P(2) + 0,5 = 1,6 --> P(2) = 0,3

But these results seem to differ. So it seems I clearly miss some important rule!

Must I not "plug in" results into other rows, as the "Gaussian" system seems to avoid? But how can there be a difference / can I be forced to avoid "plugging in" results?

This should normally be allowed, shouldn't it? Can you help me find my blind spot?

Thanks for your help in advance.

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    I want to double-check a few things. I'm used to point-decimal notation, i.e. 1.5. But when you write things like 0,25 - that's a decimal, right? Second, when you write P(1), P(2), and P(3), do you just mean separate variables that you are solving for?2011-07-10
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    One of the problems could be notation. Maybe $P(1)$, $P(2)$, $P(3)$ look too much like each other, particularly when there are numbers floating around. I have a feeling that the error rate might go down if they were called $x$, $y$, and $z$.2011-07-10
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    user6312 you are basically right, but we have to use this notation! :( But furthermore I got the problem, that I never come down to one variable! I also had this problem with later exercises. I can calculate around for 30 minutes and don't come to an end / silver lining on the horizon!2011-07-12

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