I know that linear transformations are isomorphic to matrices, in the sense that a matrix can represent a single linear transformation and vice-versa, and they share properties and operations (like composition).
But I think that matrices are also isomorphic to systems of linear equations. See, a system of linear equations can be represented by a matrix. Two equivalent linear systems are represented by two matrices that are line-equivalent. You can convert a linear system into a matrix, than operate on that matrix by applying line-operations, and then convert the resulting matrix back to a linear system which is equivalent (has the same solution) as the original one.
So, are matrices isomorphic to systems of linear equations?
This is a good question.
In an earlier question you asked about a formal specification for a vector space, so you clearly understand that mathematical objects need formal specifications. When you have two objects of the same type (for example, two vector spaces) you can ask whether they are isomorphic. That term itself has a formal specification, involving a bijection.
You are trying to use the word to describe the fact that things you do when studying linear systems can be written using the language of matrices. There is indeed a bijective correspondence that matches a linear system with its matrix of coordinates. That correspondence does have the feel of an isomorphism (your instinct is correct) but it's not usually called an isomorphism. In principle you could write formal definitions for the operations on linear systems and on matrices to codify that feeling, and then use the term. But don't bother. Use matrix language to talk about linear systems (it makes things easier) and learn as you learn more linear algebra that matrices have lots of other uses too.