  Sylvester-type matrix equations have applications in areas including control
theory, neural networks, and image processing. In this paper, we establish the
necessary and sufficient conditions for the system of Sylvester-type quaternion
matrix equations to be consistent and derive an expression of its general
solution (when it is solvable). As an application, we investigate the necessary
and sufficient conditions for quaternion matrix equations to be consistent and
derive a formula for its general solution involving $\eta$-Hermicity. As a
special case, we also present the necessary and sufficient conditions for the
system of two-sided Sylvester-type quaternion matrix equations to have a
solution and derive a formula for its general solution (when it is solvable).
Finally, we present an algorithm and an example to illustrate the main results
of this paper.
