I want to ask you which field of mathematics contains following kind of problem.
Suppose that following equations are given
$\alpha\times x_{1}=C_{1}$
$\alpha\times x_{2}=C_{2}$
$\alpha\times x_{3}=C_{3}$
$\alpha\times x_{4}=C_{4}$
$\alpha\times x_{5}=C_{5}$
....
$\alpha\times x_{n}=C_{n}$
$\alpha$ and $x_{i}$ are unknown. All $\alpha$, $x_{i}$, and $C_{i}$ are complex number. $\alpha$ is same for all equations. We know the value of $C_{i}$.
I want to know if we can correctly guess (or estimate) $\alpha$.
Is there any relationship between the estimation probability and the number of such equations. That is, if the number of equations increases, does the probability of correctly guessing $\alpha$ also increases or decreases?
Thanks.
At first glance, this looks like there is no way of determining $\alpha$ at all since the $x_i$ are unknown. If any value for $\alpha$ is chosen, then the $C_i$ determines each $x_i$ from there. Is there any other parameters to the system?