I have this linear system that follows $$ A x = b$$ where the coefficient matrix A is a square $3\times 3$ matrix defined as $$ A= \begin{bmatrix} 1 & (G-b_1) & C_1(G-b_1) \\ 1 & (G-b_2)& C_2(G-b_2) \\ 1 & (G-b_3)& C_3(G-b_3) \end{bmatrix} $$ and $$ b= \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix} $$ and the column of unknowns is $$ x= \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} $$
$G$ is a constant and I have full control over the $C$ parameters, i.e., I can change $C_1$, $C_2$ and $C_3$. However, the problem is when I choose the values of $C_1$, $C_2$ and $C_3$ to be close or reasonable, what I get is $$ \operatorname{rank}[A]=\operatorname{rank}[A|b] = 2 \hspace{.1in}\text{which is}< 3\hspace{.1in}\text{(number of unknowns)} $$
and when either one of them for example, $C1$ is too high compared to the other too $$ \operatorname{rank}[A]=\operatorname{rank}[A|b] = 3 $$
but the solution is not correct. Any suggestions? If the equations are linearly dependent, are there any techniques to make them independent to be able to solve for the three unknowns? Thanks!