$\left\{\begin{array}{l}2 x+3 y=13 \\ 3 x-5 y=10\end{array}\right.$

Step 1 ：We are given the system of equations: \[\begin{cases} 2x + 3y = 13 \ 3x - 5y = 10 \end{cases}\]

Step 2 ：We can represent this system of equations in matrix form as \[\begin{bmatrix} 2 & 3 \ 3 & -5 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 13 \ 10 \end{bmatrix}\]

Step 3 ：Let's denote the matrix of coefficients as A, the matrix of variables as X and the matrix of constants as B. So, we have \[AX = B\]

Step 4 ：To find X, we can multiply both sides of the equation by the inverse of A, if it exists. So, we have \[X = A^{-1}B\]

Step 5 ：By calculating, we find that the solution to the system of equations is \[X = \begin{bmatrix} 5 \ 1 \end{bmatrix}\]

Step 6 ：So, the solution to the system of equations is \[\boxed{x = 5, y = 1}\]