1. **State the problem:** We need to find the solution point $S$ where the two lines given by the system of equations intersect: $$\begin{cases} y = -4x + 2 \\ y = 3x - 5 \end{cases}$$ 2. **Set the equations equal to find the intersection:** Since both equal $y$, set the right sides equal: $$-4x + 2 = 3x - 5$$ 3. **Solve for $x$:** $$-4x + 2 = 3x - 5$$ $$2 + 5 = 3x + 4x$$ $$7 = 7x$$ $$x = \cancel{\frac{7}{7}}1$$ 4. **Substitute $x=1$ into one of the original equations to find $y$:** Using $y = 3x - 5$: $$y = 3(1) - 5 = 3 - 5 = -2$$ 5. **Conclusion:** The solution point $S$ is at $(1, -2)$. This matches option D, where the lines intersect at $S(1, -2)$ in the bottom-right position.