Why can we vary parameters in linear regression?

Question

I want to understand what the formal justification is for relating the response variable to new values of explanatory variables in linear regression. I see linear regression models defined as $Y=X\beta +\epsilon$, where $Y$ is a random vector, $X$ is a given matrix of constants, $\beta$ is a vector of unobservable fixed parameters, and $\epsilon$ is a random vector of errors with mean 0.

What in this formulation implies that we can generalize the relationship between $Y$ and the given rows of $X$ to "new rows" or observations? I basically understand intuitively that we're approximating the relationship between $Y$ and a some "quantity" $x$ with a linear function of $x$, but where is the idea that we can vary $x$ built into the concept of linear regression?

Answer 1

The $X$'s that you refer to as constants are values of the explanatory variables. Linear regression assumes that the outcome variable $Y$ is a linear (in the betas) function of the $X$'s. But the result is subject to a small amount of random noise which is represented by the epsilons in the model.