Question: Solve two equations in 3 variables by var1 and var2 and then by var 2 and var3

I have two equations with 3 variables(or perhaps n equations with n+1 variables) eq1:=0=c1*A1+c2*B1+c3*C1; eq2:=0=c4*A1+c5*B1+c6*C1; I can solve by e.g. solve({eq1,eq2},{A1,B1}) or any other combination of two of the variables. But if I follow the above by solve({eq1,eq2},{B1,C1}) I get then get an error. How do I find C1 as a function of A1? There must surely be a very way simple to do this. Did check the help pages but must have missed it if it is there.
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