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The subject matter of this work are the linear, three variable diophantineequation ax+by+cz=d (1), and the diophantine system ax+by+cz=d (2) ex+fy+gz=hwith the coefficients a,b,c,d,e,f,g,h being integers. Introductory numbertheory books, typically contain only a brief outline of how to solve equation(1). Even less or no material is offered on the system (2). The purpose of thiswork is to fill this gap. After some preliminary, introductory material, whichincludes the general solution of the two variable linear diophantine equationax+by=c(material which we use later in the paper); we present a complete anddetailed analysis of equation (1). We determine the precise conditions that thecoefficients a,b,c,d must satisfy in order for integer solutions to exist. Wethen derive a two-parameter, parametric description of The solution set. Thesolution set of (1), if not empty, consists of all integer triples (x,y,z) thatsatisfy (1). Geometrically, this is interpreted as the set of all latticepoints in 3-D space which lie on the plane with equation (1). Similarly, weoffer an exhaustive analysis of the system (2) by determining the preciseconditions that the coeffients must satisfy in order for integer solutions toexist. We offer seven examples with detailed solutions.

Lattice points on the plane ax+by+cz=d and the diophantine system ax+by+cz=d ex+fy+gz=h