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How do I plot a multiple regression graph? Shall it be a many dimensional plane(equals to number of explanatory variable)? Or will be the line for each beta, since the slope or beta is the partial derivative of y(explained) with respect to that particular x(explanatory)??

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In principle it's a plane for 2 independent variables or a hyperplane for more. An exception is when your independent variables are a continous and a dummy variable. In that case the dummy basically changes to the intercept only so you could show two lines with different intercepts.

A work around if you have 2 continuous independent variables is to show the relation between the dependent and one independent for several values of the other independent.

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