Suppose we are given the system of equations

$${\alpha}_{1}A+{\beta}_{1}B+{\gamma}_{1}C=x$$

$${\alpha}_{2}A+{\beta}_{2}B+{\gamma}_{2}C+{\theta}_{2}D=y$$

$${\alpha}_{3}A+{\beta}_{3}B+{\gamma}_{3}C+{\theta}_{3}D=z$$

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 $A,B,C,D$.

Is it possible to use the equations above to have a unique solution for $A,B,C$?

$${\alpha}_{1}A+{\beta}_{1}B+{\gamma}_{1}C=x$$

$${\alpha}_{2}A+{\beta}_{2}B+{\gamma}_{2}C+{\theta}_{2}D=y$$

$${\alpha}_{3}A+{\beta}_{3}B+{\gamma}_{3}C+{\theta}_{3}D=z$$

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 $A,B,C,D$.

Is it possible to use the equations above to have a unique solution for $A,B,C$?