In this paper, under certain conditions, we prove the existence of a unique solution of mixed integral equation of the second kind (MIESK) with singular kernel in the space  using Banach fixed point theorem. Then, using a numerical method, we have a linear system of Fredholm integral equations (LSFIEs). Moreover, we use a Toeplitz matrix method to obtain a linear algebraic system (LAS) which can be solved, numerically. The error estimate is discussed. Some examples are solved numerically.

mixed integral equation (MIE), linear system of Fredholm integral equations (LSFIEs), linear algebraic system (LAS), singular kernel, Toeplitz matrix method.