
System of equations
Can somebody please help me explain why the following system of equations (with variables a, b, c, d and e):
$\displaystyle 20a + 16c + 4e = 2300$
$\displaystyle 200b + 152c + 20d + 28e = 2300$
$\displaystyle 16a + 152b + 168c = 1800$
$\displaystyle 20b + 20d = 700$
$\displaystyle 4a + 28b + 32e = 500$
has no solutions? I'll put it into an augmented matrix to make it easier for you (if that helps):
$\displaystyle \begin{bmatrix}
20 & 0 & 16 & 0 & 4 & 2300 \\
0 & 200 & 152 & 20 & 28 & 2300 \\
16 & 152 & 168 & 0 & 0 & 1800 \\
0 & 20 & 0 & 20 & 0 & 700 \\
4 & 28 & 0 & 0 & 32 & 500
\end{bmatrix}$
I seem to be having trouble explaining this so if you could help me out, that would be greatly appreciated.
Thanks.

I tried cancelling it down into a more silky and manageable matrix:
200b+152c+20d+7(230020a16c)=2300
16a+152b+168c=1800
20b+20d=700 (This can be cancelled to b+d=35)
4a+28b+8(230020a16c)=500
200b+40c+20d140a=13800 <This equation has too many variables.
152b+168c+16a=1800
20b+20d=700
28b128c156a=17900
200b+40c+(70020b)140a=13800
180b+40c140a=13100<much better!(Clapping)
We now have:
180b+40c140a=13100
152b+168c+16a=700
28b128c156a=17900
In matrix form this is:
http://i116.photobucket.com/albums/o...sproblem65.jpg
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!(Surprised))
2723840+179204717440+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!(Rock)