Possible error in user guide? "statistically equivalent"

In the linear regression portion of the Stan user guide, it states that the following are “statistically equivalent”. Could someone please clarify why this is the case?

model {
  y ~ normal(x * beta + alpha, sigma);  // likelihood


model {
  for (n in 1:N) {
    y[n] ~ normal(x[n] * beta, sigma);

How does the alpha get incorporated into beta and x[n] in the second model code block?


This could certainly be clearer. The ambiguity is due to the preceding sentence:

 It would be possible to include a column of ones in the data matrix x to remove the alpha parameter.

Does that make sense?

Yes, that makes sense. Sorry, I missed that statement about redefining x and beta. I should have paid closer attention.