1. **State the problem:** Solve the system of equations using the elimination method: $$4x + y = 9$$ $$x + \frac{1}{9}y = 4$$ 2. **Goal:** Eliminate one variable by making the coefficients of either $x$ or $y$ the same in both equations. 3. Multiply the second equation by 9 to clear the fraction: $$9 \times \left(x + \frac{1}{9}y\right) = 9 \times 4$$ $$9x + y = 36$$ 4. Now the system is: $$4x + y = 9$$ $$9x + y = 36$$ 5. Subtract the first equation from the second to eliminate $y$: $$(9x + y) - (4x + y) = 36 - 9$$ $$9x - 4x + y - y = 27$$ $$5x = 27$$ 6. Solve for $x$: $$x = \frac{27}{5} = 5.4$$ 7. Substitute $x = 5.4$ into the first equation to find $y$: $$4(5.4) + y = 9$$ $$21.6 + y = 9$$ $$y = 9 - 21.6 = -12.6$$ **Final answer:** $$x = 5.4, \quad y = -12.6$$