This thesis provides an algorithm for finding the general solutions of a given system of linear Diophantine equations. A linear Diophantine equation is a polynomial equation (in any number of unknowns) with degree one and whose solutions in integers are to be determined. The concepts used in this paper are basically from Number Theory and Linear Algebra.The linear Diophantine equation of the form y1c1 + y2c2 + y3c3 + ... + yncn = e and the systems of linear Diophantine equations of the formy1c11 + y2c12 + y3c19 + ... + ync1n = 31y1c21 + y2c22 + y3c23 + ... + ync2n = e2y1c31 + y2c32 + y3c33 + ... + ync3n = e3: : : :y1cm1 + y2cm2 + y3cm3 + ... + yncmn = em where c i j, ej are given integers, for i = 1,2, ..., m and j = 1,2, ..., n were considered in this paper. The algorithm used by Stanley Kertzner in his article entitled The Linear Diophantine Equation published in American Mathematical Monthly, on March 1981 was the basis for this paper since his method of finding the general solutions of the systems of linear Diophantine equations generates all the possible solutions to the system of equations
