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.
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 $h$.