Abstract

This study aimed to investigate a generalized method for finding one integer solution to a linear equation in two variables of the form r_1 x + r_2 y = c. In school mathematics, solving linear equations in two variables is an important topic, and in number theory this problem is closely related to linear Diophantine equations. However, textbooks do not usually present a systematic method for obtaining a particular integer solution. To address this limitation, this study focused on the case where (r_1,r_2) = g and g | c, and explored a generalized formula based on the simple continued fraction expansion of r1/r2. First, cases with n = 0,1,2,3 were analyzed inductively to identify a recurring pattern. Then, this pattern was generalized into a recursive formula for generating one integer solution and justified by mathematical induction. The study also showed that the procedure can be explained structurally through matrix representation. The findings indicate that one integer solution can be obtained systematically from the coefficients of the continued fraction, and that the method can be applied to both middle school textbook problems and university-level number theory examples. Thus, this study provides a meaningful link between school mathematics and number theory.