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Math Help - System of equations

  1. #1
    Junior Member
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    System of equations

    Can somebody please help me explain why the following system of equations (with variables a, b, c, d and e):

    20a + 16c + 4e = 2300

    200b + 152c + 20d + 28e = -2300

    16a + 152b + 168c = 1800

    20b + 20d = -700

    4a + 28b + 32e = 500

    has no solutions? I'll put it into an augmented matrix to make it easier for you (if that helps):

    \begin{bmatrix}<br />
  20 & 0 & 16 & 0 & 4 & 2300 \\<br />
  0 & 200 & 152 & 20 & 28 &  -2300 \\<br />
16 & 152 & 168 & 0 & 0 & 1800 \\<br />
0 & 20 & 0 & 20 & 0 & -700 \\<br />
4 & 28 & 0 & 0 & 32 & 500<br />
\end{bmatrix}

    I seem to be having trouble explaining this so if you could help me out, that would be greatly appreciated.

    Thanks.
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  2. #2
    Super Member Showcase_22's Avatar
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    I tried cancelling it down into a more silky and manageable matrix:

    200b+152c+20d+7(2300-20a-16c)=-2300
    16a+152b+168c=1800
    20b+20d=-700 (This can be cancelled to b+d=-35)
    4a+28b+8(2300-20a-16c)=500

    200b+40c+20d-140a=-13800 <------This equation has too many variables.
    152b+168c+16a=1800
    20b+20d=-700
    28b-128c-156a=-17900

    200b+40c+(-700-20b)-140a=-13800
    180b+40c-140a=-13100<------much better!

    We now have:

    180b+40c-140a=-13100
    152b+168c+16a=-700
    28b-128c-156a=-17900

    In matrix form this is:



    The next step is finding the determinant:

    (I would write down how I got these numbers, but even Microsoft Word Equation Drawer has it's limits!)

    2723840+17920-4717440+948480+658560+368640=0

    This is only a bit of the proof. Since the determinant=0, there is either no inverse (and hence no solutions) or all the points are mapped to a line and it has planes as solutions. None of the lines are multiples of each other so they're all different. It as been a while since i've done matrices so I can't actually remember if that has any significance. :s

    I'm not really sure where else to go with this so I hope you (or another poster) can figure it out!
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