How would you expand a simple model into a multiple one? I know you are guessing it, by adding more slopes. This is actually the right answer, even if its implications are not that trivial.
We formally define a multivariate linear model as:
But what is the actual meaning of this formula? We know that the meaning for the univariate was the relationship between an increase of x and an increase of y, but what is the meaning now that we are dealing with multiple variables?
Once we adopt the ordinary least squares (OLS) again to estimate those coefficients, it turns out that it means how an increase in the given variable influences the level of y, keeping all other variables constant. We therefore, are not dealing with a dynamic model able to express the level of influence of each variable taking into consideration the level of other variables. For the sake of completeness, I have to tell you that it would also be possible to add interactions...