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My question is not about statistical programming. We know how to code the software in a regression with many independent variables. My question is about how the computer software controls for all variables at the same time mathematically.
Simple correlation of two variables is easy to understand, as each variable has the other one's full attention. In multiple regression, however, multiple independent variables are "looking at" the dependent variable at the same time, and each one looks to "subtract out" or explain the variation, but they can't all be in operation with the dependent variable at the same time. If a single given independent variable is given priority to be the first to interact with the dependent variable, and it "takes away" its share of the dependent variable, then what it takes away won't be rightfully eligible to be explained by other independent variables, as it justly should.
My intuitive sense of what happens must be mistaken, and I need an answer that does not involve formulas. The challenge for the teaching profession is how to explain complex ideas narratively or metaphorically for analysts that don't have a fast, intuitive sense for symbolic explanation. A related question is this: is the process iterative in that initial approximations (of betas) are made, and they incrementally adjusted as each of the other independent variables add more information to the initial beta estimates?