The purpose of this article is to suggest a method of estimating parameters of linear regressions containing two independent variables, when data is missing among these variables. The problem envisaged concerns the case where: (1) the independent variables are considered as fixed numbers; (2) each observation contains the values of the dependent variable and at least one of the independent variables; (3) some observations are complete. In contrast with other approaches dealing with similar problems, the technique developed in this article has the following advantages: (1) it is based on rather unrestrictive hypotheses; (2) the resulting estimators are consistent; (3) the asymptotic variances of these estimators are smaller than those of comparable estimators described in the literature. Although the question is not examined in the present article, if seems also that the proposed method offers good possibilities of generalization.