.
Let me reformulate the problem of the post in a Mathematical way:
Find the set of all solutions to the equation x + by = y,
where y is an arbitrary real number.
Solution
The given equation is equivalent to x = y - by, or, which is the same
y*(1-b) = x.
If b = 1, then x= 0;
If b=/= 1, then x = y*(1-b).
Answer. In terms of pairs (b,x), the set of solution is the set { (b,x) | x= y*(1-b) }, for any given real number y.
