1. We are asked to solve the linear system using substitution: $$2x - 4y = 7$$ $$4x + y = 5$$ 2. From the second equation, solve for $y$: $$y = 5 - 4x$$ 3. Substitute $y = 5 - 4x$ into the first equation: $$2x - 4(5 - 4x) = 7$$ 4. Distribute the $-4$: $$2x - 20 + 16x = 7$$ 5. Combine like terms: $$18x - 20 = 7$$ 6. Add 20 to both sides: $$18x - \cancel{20} + 20 = 7 + 20$$ $$18x = 27$$ 7. Divide both sides by 18: $$\frac{\cancel{18}x}{\cancel{18}} = \frac{27}{18}$$ $$x = \frac{27}{18} = \frac{3}{2}$$ 8. Substitute $x = \frac{3}{2}$ back into $y = 5 - 4x$: $$y = 5 - 4 \times \frac{3}{2} = 5 - 6 = -1$$ 9. Final solution: $$\boxed{\left( \frac{3}{2}, -1 \right)}$$