1. **State the problem:** Solve the system of linear equations: $$0.4x + 0.5y = 0.2$$ $$0.3x - 0.8y = 2.5$$ 2. **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 by 4 to align coefficients of $x$: $$3(0.4x + 0.5y) = 3(0.2) \Rightarrow 1.2x + 1.5y = 0.6$$ $$4(0.3x - 0.8y) = 4(2.5) \Rightarrow 1.2x - 3.2y = 10$$ 4. **Subtract the second from the first to eliminate $x$:** $$ (1.2x + 1.5y) - (1.2x - 3.2y) = 0.6 - 10 $$ $$ \cancel{1.2x} + 1.5y - \cancel{1.2x} + 3.2y = -9.4 $$ $$ 4.7y = -9.4 $$ 5. **Solve for $y$:** $$ y = \frac{-9.4}{4.7} = -2 $$ 6. **Substitute $y = -2$ into the first original equation:** $$ 0.4x + 0.5(-2) = 0.2 $$ $$ 0.4x - 1 = 0.2 $$ $$ 0.4x = 1.2 $$ 7. **Solve for $x$:** $$ x = \frac{1.2}{0.4} = 3 $$ **Final answer:** $$ x = 3, \quad y = -2 $$