1. The problem is to determine how many solutions the system of linear equations has: $$4x + 9y = -2$$ $$11x - 17y = -19$$ 2. To find the number of solutions, we check if the lines represented by these equations are parallel, coincident, or intersecting. 3. The general form of a linear equation is $Ax + By = C$. For the first equation, $A_1=4$, $B_1=9$, $C_1=-2$. For the second, $A_2=11$, $B_2=-17$, $C_2=-19$. 4. Calculate the ratios: $$\frac{A_1}{A_2} = \frac{4}{11}$$ $$\frac{B_1}{B_2} = \frac{9}{-17} = -\frac{9}{17}$$ $$\frac{C_1}{C_2} = \frac{-2}{-19} = \frac{2}{19}$$ 5. Since $\frac{A_1}{A_2} \neq \frac{B_1}{B_2}$, the lines are not parallel. 6. When the ratios of $A$ and $B$ are not equal, the system has exactly one solution (the lines intersect at a single point). Final answer: one solution.