This paper is mainly devoted to exact solutions to the initial value problem for linear conformable systems with variable coefficients. The famous method known as the generalized Peano-Baker series, which inholds the conformable integral, is exploited to acquire the state-transition matrix. A representation of an exact solution in a closed interval for linear confromable systems with variable coefficients is determined with the help of this matrix. It is verified by showing that the determined exact solution satisfies the systems step by step. Moreover, another exact solution in the same closed interval is identified thanks to the method of variation of parameters. The existence and uniqueness of the second exact solution to the systems are provided by the Banach contraction mapping principle. This provides that the representations of the two solutions coincide although they are obtained by completely different approaches and they have completely different structures. A couple of examples are presented to exmplify the use of the findings.