1. **State the problem:** We are given the system of equations: $$-7x + 2y = 12$$ $$-x - 3y = 5$$ We need to find the value of the determinant $D$ for the coefficient matrix of this system. 2. **Identify the coefficient matrix:** The coefficient matrix is formed by the coefficients of $x$ and $y$ in both equations: $$A = \begin{bmatrix}-7 & 2 \\ -1 & -3\end{bmatrix}$$ 3. **Calculate the determinant $D$:** The determinant of a $2 \times 2$ matrix $\begin{bmatrix}a & b \\ c & d\end{bmatrix}$ is calculated as: $$D = ad - bc$$ For our matrix: $$D = (-7)(-3) - (2)(-1)$$ $$D = 21 + 2$$ $$D = 23$$ 4. **Final answer:** The value of the determinant $D$ is: $$\boxed{23}$$