Why does multiple regression equation make sense? the equation of multiple regression here can be interpreted as y = b1x1 + b2x2 + … + bnxn + c.
where bn is the slope of the regression , xn is the independent variable
I understand that if we were to do the simple regression which involves like 2 independent variables we can calculate a line with a certain slope at a specific y-intercept to be as close as as possible from the training data we have on the graph and having the least amount of errors. From what I heard Multiple regression is an extension of simple regression but I can't quite grasp Multiple regression equation. Can someone explain it's intuition?
 A: Intuitively you can think of multiple regression as extending the two-dimensional "draw a line through the cloud of points" into 3 or more dimensions.
So in a simple linear regression with one independent variable (X) and one dependent variable Y, we can imagine an X-Y plot, with a Y axis and an X axis. Each observation in the dataset is a point (x,y) somewhere in that plane. If we plot them all they create a cloud. We draw a line
$y=\beta_0+\beta_1x$
through that line, and we use some matrix algebra to find the specific values of $\beta_0$ (the intercept) and $\beta_0$ (the slope) that minimize the sum of the squared errors between the Y values predicted by the line and the actual Y values of the point.
If we wanted to add a second independent variable Z to the equation then now we have a three dimensional space, with X, Y and Z coordinates. We can still plot each observation in that space, producing a 3D cloud of points. Now our goal is to find the two dimensional plane that goes through the "center" of the cloud. Of course a plane floating in 3d space doesn't have just one slope: it has two partial slopes, one (call it $\beta_1$) showing how much the plane tilts in the Y direction when Z stays constant and X increases, and one (call it $\beta_2$) showing how much Y increases when Z goes up by one and X stays the same. The equation of that plane is
$y=\beta_0+\beta_1x+\beta_2z$
This is a lot harder to visualize but we can use the same matrix algebra to find the "best" values for the various beta coefficients (the ones that minimize the sum of squared errors). And even if we add more variables, going into 4 or more dimensions, the matrix algebra still works, even though it's utterly impossible to visualize.
So this is one way to think about what's going on. Multiple regression gives you a bunch of partial slopes that tell you how much we think Y will increase if you increase a particular variable by one unit, but hold everything else constant.
