1. **State the problem:** Solve the system of linear equations using the elimination method: $$\begin{cases} 4x - y = 1 \\ 2x + 3y = 11 \end{cases}$$ 2. **Explain the elimination method:** We aim to eliminate one variable by adding or subtracting multiples of the equations. 3. **Multiply the second equation by -2 to align coefficients of $x$:** $$-2(2x + 3y = 11) \Rightarrow -4x - 6y = -22$$ 4. **Add the first equation and the new equation:** $$\begin{aligned} &(4x - y) + (-4x - 6y) = 1 + (-22) \\ &\cancel{4x} - y - \cancel{4x} - 6y = -21 \\ &-7y = -21 \end{aligned}$$ 5. **Solve for $y$:** $$y = \frac{-21}{-7} = 3$$ 6. **Substitute $y=3$ into the first equation to find $x$:** $$4x - 3 = 1$$ $$4x = 4$$ $$x = 1$$ 7. **Final solution:** $$\boxed{(x,y) = (1,3)}$$