I have come across an excellent answer of @Gung to a previous post. However, I still have a doubt. Using the example of @Gung, suppose that there are three main variables: the dependent variable Y, and two main predictors, X1 and X2. But let X1 and X2 be mutually exclusive dummies, ie, if X1 (X2) is 1, X2 (X1) is 0 (and to avoid the dummy trap let there be also cases in which X1 and X2 are both 0).
In this case, to study the effect of X1 over Y, is "controlling" for X2 similar to "ignoring" it?
Moreover, to analyze the effect of X1 and X2 over Y, is this model
Y = a + b * X1 + c * X2
interchangeable with a two-equations system like this,
(1) Y = v + w * X1, (2) Y = h + k * X2?