Adding a redundant equation to a system of algebraic equation that generates inconsistent output

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In a dynamics problem I am working on, two different systems of equation that describe the motion of a body are both correct and should generate the same solutions.
One of the systems of equations is eq1, eq2, and eq3 and the other is eq2, eq3, eq4. When they are considered separately, sol is equated to sol = solve([eq1 eq2 eq3], [b c N]); or sol = solve([eq2 eq3 eq4], [b c N]); and c has the same value in each case. The same solution should be obtained if a redundant equation, either eq4 or eq1, is added, but that is not the case, as in this case the solution is the empty set. Can anyone explain why?
syms a b c N theta m L b g
eq1 = N-m*g == m*L/2*(b*sin(theta)+a^2*cos(theta)) ;
eq2 = -m*g*L/2*sin(theta) == 1/3*m*L^2*(b)+L/2*m*cos(theta)*c;
eq3 = c == L/2*(-b*cos(theta)+a^2*sin(theta)) ;
eq4 = -N*L/2*sin(theta) == 1/12*m*L^2*(b);
sol = solve([eq1 eq2 eq3 eq4], [b c N]); %[a b c] = theta' theta'' x''
sol_a = simplify(sol.c)
sol2 = sol.c

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