If X=Y+Z, Is it ever useful to regress X on Y? If we have X and Y that are mathematically dependent: X = Y + Z, is it 'forbidden' to use Y as a predictor to X in linear regression?  I'm trying to find a concise explanation for why it is, or isn't.
One explanation I've found is that you shouldn't use simple linear regression in this case, but multiple linear regression, with Y and Z as predictors, but this doesn't seem right. The mathematical relationship is exact, and the multiple regression would return coefficients 0, 1, and 1  for X = 0 + 1Y + 1Z. So, there is something more fundamental I'm missing.
The purpose for linear regression is predicting the value of one variable from the other. But we already know that X can be calculated as the sum of Y and Z, so effectively, we are calculating the regression between Y and Y + Z. Is this 'forbidden'?
But, if we can only measure Y, does the regression give us a tool to predict X? Isn't it better to find the regression between Y and Z? I'm confused.

Two people step on a scale. The scale outputs only the total weight of the couple.
If you know the weight of the first person ($Y$) can you guess what the scale will output ($X$) without knowing anything about the weight of the second person ($Z$)?
Intuitively, there has to be some connection. If $Y$ is very heavy, it is likely that the total weight $X$ will also be high.
Regressing X on Y here is just trying to put a number to that connection by looking at a bunch of real-life examples of total weight $X$ given first person weight $Y$.
It is not forbidden, as you say. It is simply trying to make the best guess given the information you have.
Of course if you know the weight of both people $Y$ and $Z$ and you know how scales work, you don't need to make observations and regression.
