1. **State the problem:** Solve the system of equations $$\begin{cases} -x + 4y = 6 \\ 2x - y = -5 \end{cases}$$ 2. **Rewrite each equation to express $y$ in terms of $x$: ** From the first equation: $$-x + 4y = 6 \implies 4y = x + 6 \implies y = \frac{x + 6}{4}$$ From the second equation: $$2x - y = -5 \implies -y = -5 - 2x \implies y = 2x + 5$$ 3. **Set the two expressions for $y$ equal to find $x$: ** $$\frac{x + 6}{4} = 2x + 5$$ Multiply both sides by 4 to clear the denominator: $$x + 6 = 4(2x + 5)$$ $$x + 6 = 8x + 20$$ 4. **Isolate $x$: ** $$x + 6 = 8x + 20$$ $$6 - 20 = 8x - x$$ $$-14 = 7x$$ $$x = \frac{-14}{7} = -2$$ 5. **Substitute $x = -2$ back into one of the equations to find $y$: ** Using $y = 2x + 5$: $$y = 2(-2) + 5 = -4 + 5 = 1$$ 6. **Solution:** The system's solution is $$\boxed{(-2, 1)}$$ This matches the point where the two lines intersect on the graph.