1. **State the problem:** Solve the system of linear equations: $$6x - 11y = -83.8$$ $$5x + 3y = -9$$ 2. **Choose a method:** We will use the elimination method to solve for $x$ and $y$. 3. **Eliminate one variable:** Multiply the first equation by 3 and the second equation by 11 to align coefficients of $y$: $$3(6x - 11y) = 3(-83.8) \Rightarrow 18x - 33y = -251.4$$ $$11(5x + 3y) = 11(-9) \Rightarrow 55x + 33y = -99$$ 4. **Add the two equations:** $$18x - 33y + 55x + 33y = -251.4 - 99$$ $$73x = -350.4$$ 5. **Solve for $x$:** $$x = \frac{-350.4}{73} = -4.8$$ 6. **Substitute $x$ back into one original equation:** Use $5x + 3y = -9$: $$5(-4.8) + 3y = -9$$ $$-24 + 3y = -9$$ $$3y = -9 + 24 = 15$$ 7. **Solve for $y$:** $$y = \frac{15}{3} = 5$$ **Final answer:** $$x = -4.8, \quad y = 5$$