I'm new to much of this and find that a good way to wrap my head around some of these concepts is to calculate them by hand. After working through linear regression, I thought multiple regression would be straightforward because I read that multiple regression is the linear combination of the independent variables. But I can't seem to learn how to calculate the y-intercept ($b_0$) for multiple regression. Maybe I'm using the wrong terminology because I'm finding this difficult to google.
For simple linear regression, the y-intercept is:
$b_0 = \bar y -b_1\bar x$
The equation for multiple regression is:
$ \hat y = b_0 + b_1x_1 + b_2x_2 +b_3x_3... $
I don't understand where the $b_0$ comes from in the multiple regression equation because wouldn't we have a different $b_0$ for every independent variable? How do they turn into a single y-intercept?
Thanks in advance!