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?
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