Linear System F0Efae

1. **Stating the problem:** Solve the system of linear equations: $$x - 5y = -8$$ $$3x - 2y = -11$$ 2. **Formula and method:** We use the matrix method to solve the system. The system can be written as: $$\begin{pmatrix} 1 & -5 \\ 3 & -2 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -8 \\ -11 \end{pmatrix}$$ 3. **Calculate the determinant:** $$\det = 1 \times (-2) - (-5) \times 3 = -2 + 15 = 13$$ 4. **Find the inverse matrix:** $$\frac{1}{13} \begin{pmatrix} -2 & 5 \\ -3 & 1 \end{pmatrix}$$ 5. **Multiply inverse matrix by constants vector:** $$\begin{pmatrix} x \\ y \end{pmatrix} = \frac{1}{13} \begin{pmatrix} -2 & 5 \\ -3 & 1 \end{pmatrix} \begin{pmatrix} -8 \\ -11 \end{pmatrix}$$ 6. **Calculate each component:** $$x = \frac{1}{13}((-2)(-8) + 5(-11)) = \frac{1}{13}(16 - 55) = \frac{-39}{13} = -3$$ $$y = \frac{1}{13}((-3)(-8) + 1(-11)) = \frac{1}{13}(24 - 11) = \frac{13}{13} = 1$$ 7. **Final answer:** $$x = -3, \quad y = 1$$ This means the solution to the system is $x = -3$ and $y = 1$.