For the following
$$Y = B_0 + B_1X_1 + B_2X_2,$$
when will $B_1$ be equal to the correlation between $Y$ and $X_1$?
When will $B_2$ be equal to the correlation between $Y$ and $X_2$?
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The answer to both questions is that a slope coefficient will equal the correlation value when you have the equation in standardized form (i.e. when the Y and X variables are z-scores) and when the predictors are independent of each other.
Not a full proof of what I said - but you can see the two-predictor case here and you'll observe that the standardized coefficient will equal the correlation between Y and Xi when the correlation between X1 and X2 is r = 0