I am trying to fit a multiple regression model to my data. I am testing the hypothesis that outcome Y is linearly related to independent variable X, while controlling for a linear relationship with covariate C, such that: Y ~ X + C

I have STRONG evidence that the true relationship describing Y ~ X does go through the origin, so without the covariate C I would force the model through the origin.

However, I have no reason to believe that the relationship describing Y ~ C goes through the origin.

So, is it valid to force the model through the origin point? (my hunch is no but I'd like to hear others' thoughts). X is the variable I'm interested in here. Thanks!