Determine whether a system of equations implies the equality of two variables

I have a system of equations of the form $a_i\alpha+b_i\beta+c_i\gamma+d_i\delta=2$. All coefficients are positive integers, and I also have the extra restriction that $0<\alpha,\beta,\gamma,\delta<1$.

My question is whether there exists a technique to easily determine whether this system implies that $\alpha=\beta$? (or any other combination of variables, but I do need to be able to ask the question for a specific pair of variables) I'm especially looking for techniques that can easily and efficiently be implemented in a computer program.

Maybe it is also best that I mention that I'm looking for cases where there is a solution if and only if those two variables are equal, and not the cases where there is a solution such that they might be equal. An example is if there is an equation with coefficients $(x,y,z,u)$ and one with coefficients $(y,x,z,u)$, such that not both $x$ and $y$ are equal to zero. These are some cases, but there are still other situations that also imply the equality, but that I'm missing if I only perform that check.

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