This paper investigates the existence theorem for solutions to hyperbolic linear systems and presents the judgment conditions for the existence of such solutions. It extends the judgment conditions applicable to general linear systems, expanding the research scope from real matrices, a focus in advanced algebra, to hyperbolic matrices. The paper derives the form of solutions to hyperbolic linear systems and conducts a more comprehensive discussion, categorizing scenarios into cases of solvability, unsolvability, and uniqueness of solutions. The conclusions drawn in this study can be further applied to research on a broader range of physical problems with concrete physical backgrounds, laying a solid research foundation and injecting new momentum into the application of hyperbolic analysis in the field of algebra.