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How do you find the value of $729a+81b+9c+d-(1000e+100f+10g+h)$ if:

\begin{cases} a+b+c+d=8e+4f+2g+h \\ 27a+9b+3c+d=64e+16f+4g+h \\ 125a+25b+5c+d=216e+36f+6g+h \\ 343a+49b+7c+d=512e+64f+8g+h+13 \end{cases}

This system looks like a system from an AIME problem, but I'm not entirely sure how to approach this because the coefficients are exponents. I am pretty sure there is a way to manipulate these such that you can get rid of many variables, such as $d$ and $h$, but I'm not sure how. A hint to how to start would be appreciated!

I'm sorry I didn't mention what I knew, but I don't really know how to work matrices. Are there any other ways?

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    Just from the shape of the problem, I would immediately assume that there is some linear combination of the four equations that gives $729a+81b+9c+d=100e+100f+10g+h$, or something damned close (the $13$ in the last equation is probably there just to give us a non-zero answer at the end), and I would begin to look for said linear combination. – 2017-01-15
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    The exponential coefficients make it hard to know which ones to combine. Any hint for where to combine or maybe get rid of the 13? Or do I just ignore that last equation? – 2017-01-15
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    clearly need $1000e$ in the title – 2017-01-15
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    sorry, my bad! You're right :) – 2017-01-15
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    https://en.wikipedia.org/wiki/Vandermonde_matrix – 2017-01-15

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The coefficients to multiply the four equations are $-1,4,-6,4$. Then you get your expression as a linear combination of the four equations, and the result is 52

I looked which linear combination of the four equations gives us the right coefficients for $a,b,c,$ and $d$, then I solved the system. Then I verified that I get the right coefficients for $e,f,g,h$ and I have 52 left from multiplying the free term (13) by 4

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    I'm sorry, but I'm confused. There are 8 coefficients in total...? Also, how'd you get there? Thanks! – 2017-01-15
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    I only worried about the LHS. Like [this](http://www.wolframalpha.com/input/?i={{1,27,125,343},{1,9,25,49},{1,3,5,7},{1,1,1,1}}^-1*{{729},{81},{9},{1}}). Then, I verified for the RHS. – 2017-01-15
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    Ok, thanks! Is this a Vandermonde Matrix? – 2017-01-15
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    Looking at it, the matrix of a system is a Vandermonde matrix. But I just solved the system using Wolfram Alpha, and I didn't worry about its structure. – 2017-01-15