Linear Diophantine equation are introduced to determine and search for integral solutions according to the associated variables. In this paper, the mathematical techniques are approached to solve Linear Diophantine equation with two, three unknowns and system of linear Diophantine equation by Euclidean algorithmic, congruence modulo n and unimodular row reduction. The modulo arithmetic operation is also applied for linear Diophantine equation in three variables. The Euclidean algorithmic is an efficient method to compute the greatest common divisor (g.c.d) of two integers. The extended Euclidean algorithmic is used also to compute the greatest common divisor (g.c.d) of two positive integers and write this greatest common divisor as an integer linear combination of two positive integers. Moreover, the mathematical methods of Diophantine equations in two and three variables are presented to practical applications in reality life such as buying and selling, traffic flux and the number of atoms of a chemical substance to investigate positive integral solutions.