Solving three-variable nonlinear equation systems

Question

A physical problem which I've been studying leads to the following nonlinear equation system to be solved: $$\alpha\cdot79\cdot A_1 +(1-\alpha)\cdot 1025 \cdot B_1 = C_{11}$$ $$\alpha\cdot145\cdot A_1 +(1-\alpha)\cdot 696 \cdot B_1 = C_{12}$$ $$\alpha\cdot12\cdot A_1 +(1-\alpha)\cdot 1578 \cdot B_1 = C_{13}$$ In these equations, $\alpha$ is an UNKNOWN universal coefficient, $A_1$ and $B_1$ are UNKNOWN variables to be obtained, and $C_{11}, C_{12}, C_{13}$ are KNOWN constants obtained from experiments. In total, we have 9 pairs of $A$ and $B$ to be obtained the corresponding experimental results $C$, for example, the pair of $A_2$ and $B_2$ should hold the following relationship with $C$: $$\alpha\cdot79\cdot A_2 +(1-\alpha)\cdot 1025 \cdot B_2 = C_{21}$$ $$\alpha\cdot145\cdot A_2 +(1-\alpha)\cdot 696 \cdot B_2 = C_{22}$$ $$\alpha\cdot12\cdot A_2 +(1-\alpha)\cdot 1578 \cdot B_2 = C_{23}$$ and the same for the other $A_n, B_n$ and $C$. But remember, the coefficient $\alpha$ is unknown. Therefore in total there are $2 \times 9 +1 = 19$ coefficients to be obtained from 27 equations. My question is, what is the best algorithm to obtain the best solution to both the universal coefficient $\alpha$ and the pairs of $A,B$ please? Thank you very much!

Answer 1

The first two equations suffice to determine $\alpha A_1$ and $(1-\alpha) B_1$. The third equation is either true or false with those values of $\alpha A_1$ and $(1-\alpha) B_1$, so (depending on how $C_{13}$ relates to $C_{11}$ and $C_{12}$) either it makes the system inconsistent or it doesn't add any new information. Similarly for the other $\alpha A_i$ and $(1-\alpha) B_i$. If those are all the constraints, you can choose $\alpha$ arbitrarily (as long as it's not $0$ or $1$), and then find $A_i$ and $B_i$.