1. **State the problem:** Solve the system of linear equations: $$\begin{cases} 2x + y = 1 \\ x - 2y = 18 \end{cases}$$ 2. **Formula and rules:** We can solve this system using substitution or elimination. Here, we'll use substitution. 3. **Isolate $y$ from the first equation:** $$y = 1 - 2x$$ 4. **Substitute $y$ into the second equation:** $$x - 2(1 - 2x) = 18$$ 5. **Simplify:** $$x - 2 + 4x = 18$$ $$5x - 2 = 18$$ 6. **Add 2 to both sides:** $$5x - \cancel{2} + \cancel{2} = 18 + 2$$ $$5x = 20$$ 7. **Divide both sides by 5:** $$\frac{5x}{\cancel{5}} = \frac{20}{\cancel{5}}$$ $$x = 4$$ 8. **Substitute $x=4$ back into $y = 1 - 2x$:** $$y = 1 - 2(4) = 1 - 8 = -7$$ 9. **Final answer:** $$\boxed{(x, y) = (4, -7)}$$