I have a deceptively simple-looking problem.
$$A + B = A'\\ C + D = B'\\ A + C = C'\\ B + D = D'$$
On LHS $4$ variables $A, B, C, D$ On RHS $4$ variables $A', B', C', D'$
Is it possible to express $A$ in terms of any of the ' variables alone? i.e find $A = f(A', B', C', D'), B = g(A', B', C', D')$, e.t.c.
Intuitively it seems there should be a relatively simple way to re-express LHS variables in terms of RHS variables. But I'm starting to think maybe that intuition is wrong?
a) If so, how do I do it?, is there some kind of general method? What is it called?
b) If not, prove it is not possible (ideally other than brute force). Is this a known/studied problem? Do you have any references? What is it called?
Thanks in advance!
Writing this in terms of matrices, you have: $$ \begin{bmatrix} 1&1&0&0\\ 0&0&1&1\\ 1&0&1&0\\ 0&1&0&1 \end{bmatrix} \begin{bmatrix}A\\B\\C\\D\end{bmatrix}=\begin{bmatrix}A'\\B'\\C'\\D'\end{bmatrix}. $$
We can write $A,\cdots,D$ in terms of $A',\cdots,D'$ iff the matrix is invertible. Since the determinant of $A$ is zero, this is not possible.
More precisely, observe that $A=B=C=D=0$ and $A=D=1$, $B=C=-1$ give the same RHS of $A'=B'=C'=D'=0$.
In other words, for any valid $A'$, $B'$, $C'$, and $D'$, there are infinitely many $A$, $B$, $C$, and $D$'s that satisfy the equations.