How could I describe how solving a system of equations using Matrices is similar and different than solving an equation like 2x + 3 = 8 to NON MATH friends?
You first have to understand linearity first off, since matrices are the most general form of linear objects.
Once you do this, you need to understand how a linear object (i.e. a matrix) transforms a vector or another linear object with certain properties (through matrix multiplication).
By explaining things in terms of rotations, scaling, and translations, you can explain how an inverse matrix "undoes" what a non-inverse matrix did.
This is the geometric idea behind a matrix (i.e. a linear operator) and depending on the operator, you can undo these things (i.e. a square matrix with non-zero determinant) and get back your original vector.