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In this paper, we provide some solvability conditions in terms of ranks forthe existence of a general solution to a system of $k$ Sylvester-typequaternion matrix equations with $3k+1$ variables$A_{i}X_{i}+Y_{i}B_{i}+C_{i}Z_{i}D_{i}+F_{i}Z_{i+1}G_{i}=E_{i},~i=\overline{1,k}$.As applications of this system, we present rank equalities as the necessary andsufficient conditions for the existence of a general solution to some systemsof quaternion matrix equations$A_{i}X_{i}+(A_{i}X_{i})_{\phi}+C_{i}Z_{i}(C_{i})_{\phi}+F_{i}Z_{i+1}(F_{i})_{\phi}=E_{i},~i=\overline{1,k}$.

A system of $k$ Sylvester-type quaternion matrix equations with $3k+1$ variables