Assume a model with 1 numerical explanatory variable (x1) and 1 categorical explanatory variable with 2 category levels (Red, Blue). Further assume the model includes the possibility that the relationship between the numerical explanatory variable and the response depends on the levels of the categorical variable (include an interaction between the numerical and categorical variable).
x2 is defined to
take the value of 1 for Red and 0 for Blue.
The population model is 
μy=β0+β1x1+β2x2+β3x1x2
The simplified version of the model for the red category is the following:
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