First, we show analytically that the nexus between Least Squares (LS) estimators of multiple and simple regression coefficients are exactly the same as between partial and total derivatives of the general function of a given number of independent variables. Second, LS estimators of multiple and simple regression coefficient vectors correspond, respectively, to net output and gross output vectors in Leontief's input/output matrix equation where the input coefficient matrix is composed of LS estimators of the coefficients in simple regression equations that can be formed from the given number of regressors. Third, we show that each element of LS estimator of multiple regression coefficient vector is represented by Cramer's rule. Finally, we show that LS estimator of each multiple regression coefficient as represented by Cramer's rule can be transformed to its counterpart in Frisch and Waugh Theorem. Thus, the fragments in regression theory are related in a unifying manner, involving the works of two Nobel laureates in Economics.