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I have eight equations with variables $\{a,b,c,d,e,f,g,h\}:$ $$a*b*c*d=64\\ b+c+d-e=6\\ a*e=8\\ b+f=6\\ b*f=8\\ c*g=8\\ d+h=9\\ d*h=8$$

I know there are 4 solutions, but is there any deterministic way of solving these? I would like to figure out the first step that naturally leads to at least one of the variables being solved for. Any help? Thank you.

aarbee
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Carl Hill
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Notice that you have two equations involving only $b$ and $f$, and two involving only $d$ and $h$. This is probably a good place to start.

For each of these mini-systems, there will be two solutions, which gives four solutions for the variables $b,d,f,$ and $h$. If there are indeed only four solutions, then the other four variables should be completely determined having chosen a solution for $b,d,f,$ and $h$.

Jared
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  • Indeed - once $b,d$ are known, we have equations for $ac$ and $ae$ which gives $e=kc$ for some constant $k$, and also for $c-e$ - this gives unique values for $a, c, e$ (provided you don't end up dividing by zero). Finally $g$ is obtained from $cg=8$ – Mark Bennet Jul 29 '13 at 18:45