I was trying to calculate the j parameters for the double perovskite structure compound La2CoMnO6 the structure and the interaction pathways are shown in the figure below, to calculate the j parameters i have used the convention H = sum(i,j), and i have only considered the nearest neighbour contributions, the equations for ferromagnetic and different antiferromagnetic spin arrangements obtained are as follows (where C1-C7 are different configurations)

Efm = 8j3+4j4+4j5+8j6

Ec1 = -8j3-4j4+4j5+8j6

Ec2 = 8j3+4j4-4j5-8j6

Ec3 = -8j3+4j4-4j5+8j6

Ec4 = 8j3-4j4+4j5-8j6

Ec5 = -8j3+4j4+4j5-8j6

Ec6 = -8j3-4j4-4j5-8j6

Ec7 = 8j3-4j4-4j5+8j6

Questions

1- If I want to solve these equations by using substitution and elimination method then these are the possible ways

j3 = 1/32*(Efm+Ec2+Ec4+Ec7), similarly other combinations

j4 = 2*j3+[1/2*(Ec3+Ec5)] or j4 = 1/32*(Efm+Ec2+Ec3+Ec5) or j4 = -2*j3-(Ec1+Ec6), similarly other combinations

j5 = 1/16*(Ec1+Ec4+Ec5-Ec6) or j5 = 2*j3-[1/2(Ec2+Ec7)] or j5 = 1/16*(Ec2-Ec4-Ec5+EC6) or 1/32*(Efm-Ec2+Ec4-Ec7), similarly other combinations

j6 = after obtaining the values of j3,j4 and j5 they can be substituted in any one of the equations and j6 can be obtained or

j6 = 1/32*(Efm+Ec1-Ec2-Ec6)-1/2*Ec5 or j6 = 1/16*(Efm+Ec3)-1/2*Ec4 or j6 = -1/32*(Ec2+Ec4+Ec5+Ec6), similarly other combinations

We can see that there is no unique way to solve for the j in that case which equation is to be considered to obtain the value of j? in other words what is the ideal method to solve the equations?