1. **State the problem:** Solve the system of linear equations using the inverse matrix method: $$\begin{cases} 2x - 3y = -1 \\ -3x + 5y = 3 \end{cases}$$ 2. **Write the system in matrix form:** $$AX = B$$ where $$A = \begin{bmatrix} 2 & -3 \\ -3 & 5 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \end{bmatrix}, \quad B = \begin{bmatrix} -1 \\ 3 \end{bmatrix}$$ 3. **Formula for the inverse of a 2x2 matrix:** For $$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix},$$ its inverse is $$A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$$ provided that the determinant $ad - bc \neq 0$. 4. **Calculate the determinant:** $$ad - bc = (2)(5) - (-3)(-3) = 10 - 9 = 1$$ Since the determinant is 1 (not zero), the matrix is invertible. 5. **Find the inverse matrix:** $$A^{-1} = \frac{1}{1} \begin{bmatrix} 5 & 3 \\ 3 & 2 \end{bmatrix} = \begin{bmatrix} 5 & 3 \\ 3 & 2 \end{bmatrix}$$ 6. **Multiply the inverse matrix by B to find X:** $$X = A^{-1}B = \begin{bmatrix} 5 & 3 \\ 3 & 2 \end{bmatrix} \begin{bmatrix} -1 \\ 3 \end{bmatrix} = \begin{bmatrix} 5(-1) + 3(3) \\ 3(-1) + 2(3) \end{bmatrix} = \begin{bmatrix} -5 + 9 \\ -3 + 6 \end{bmatrix} = \begin{bmatrix} 4 \\ 3 \end{bmatrix}$$ 7. **Interpret the solution:** The solution to the system is $x = 4$ and $y = 3$. This means the point $(4, 3)$ satisfies both equations. **Summary:** We used the inverse matrix method by first writing the system as $AX = B$, then finding $A^{-1}$ using the formula for 2x2 matrices, and finally multiplying $A^{-1}$ by $B$ to get the unique solution vector $X$.