Suppose we are given the system of equations $$\alpha_1A+\beta_1B+\gamma_1C=x$$ $$\alpha_2A+\beta_2B+\gamma_2C+\theta_2D=y$$ $$\alpha_3A+\beta_3B+\gamma_3C+\theta_3D=z$$
where $\alpha_i,\beta_i,\gamma_i,\theta_i$ are chosen from finite field $\mathbb{F}_q$ where $q$ is prime. Note that the variables here are $A, B, C, D$.
Question: Is it possible to use the equations above to have a unique solution for $A,B,C$? We can fix any conditions on the coefficients so that we can achieve the aim. It doesn't matter if $D$ cannot be found. I have tried to restrict the $\alpha, \beta$ and $\gamma$ such that the square matrix containing $\alpha,\beta,\gamma$ has rank $3$. But I don't know how to proceed from here.
Remark: If I have a $3 \times 3$ matrix which has rank $3$, then by adding a column to the matrix, what is the rank of the new matrix?
Temporarily set $D=0$. Call the resulting equations the "reduced" equations. If the determinant for (of the $\alpha,\beta,\gamma$) is non-zero, Cramer's rule will give you a unique solution for the reduced system. So you will get more than one solution to the original equations.
So the interesting case is where $\Delta=0$. That means there are either no or many solutions to the reduced equations. If there are many, then there are clearly many to the original equations.
If there are none, then we have to consider what happens in the original set. If $\theta_1=\theta_2=0$, there are still none. If $\theta_1=\theta_2\ne0$, then the original equations will have a solution provided that we do not have the last two original equations contradicting each other. but we will have more than one solution unless $q$ is small. If $\theta_1\ne\theta_2$ then we will have more than one solution.
But some care is probably needed if $q$ is really small, eg 2. Maybe in that case some of the "many"s become 1.