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We investigate the solvability of a general class of mixed Hessian type equations on compact Kähler manifolds. Based on an observation in Guan and Zhang [Pure Appl. Math. Q. 17 (2021), pp. 865–907], the mixed Hessian equations can be rewritten as elliptic and concave equations which satisfy the general framework in Székelyhidi [J. Differential Geom. 109 (2018), pp. 337–378] and Phong and Tô [Ann. Sci. Éc. Norm. Supér. (4) 54 (2021), pp. 793–829] except one condition. We notice that the absence of this condition can be overcome by using the particular structure of the mixed Hessian equations. As an application, the solvability of the deformed Hermitian Yang-Mills equations in dimension three was also studied. [ABSTRACT FROM AUTHOR]