1. ## 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):

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

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

Thanks.

2. 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!