Holding other variables constant in multiple regression

Someone asked me to explain what it means to "hold the other variables constant" in a multiple regression. One way of answering this great question is through a surface chart, a 3D representation. Below I've made a surface chart with two X independent variables, and the Y dependent variable on the vertical axis. So the height of the coloured surface shows the size of the dependent variable at any values of the two independent variables. I used the regression equation yhat = -18.36 + 2*X1 + 4.7*X2, which was Q2 in the in-class work. I calculated the value of Y for X1 and X2 from 0 to 50 and 0 to 15 respectively. Here is the plot:

and here is the data:

The Y values are in the internal cells, the two X variables along the edges. So, for example, if we held X2 constant at 3, then you can see that the change in the Y variable as X1 changes from 20 to 30 would be from 36.1 to 56.1. This is a simple example. Often the surface isn't flat but curved. HTH.