1. **State the problem:** Solve the system of linear equations: $$0.5x + 2y = 10$$ and $$y = 3x - 4.5$$ 2. **Use substitution method:** Since the second equation gives $y$ in terms of $x$, substitute $y = 3x - 4.5$ into the first equation. 3. **Substitute and simplify:** $$0.5x + 2(3x - 4.5) = 10$$ $$0.5x + 6x - 9 = 10$$ $$6.5x - 9 = 10$$ 4. **Isolate $x$:** $$6.5x = 10 + 9$$ $$6.5x = 19$$ 5. **Solve for $x$:** $$x = \frac{19}{6.5}$$ Show cancellation: $$x = \frac{19}{\cancel{6.5}} \times \frac{\cancel{1}}{1} = \frac{19}{6.5}$$ Simplify the fraction: $$x = \frac{19}{6.5} = \frac{19}{\frac{13}{2}} = 19 \times \frac{2}{13} = \frac{38}{13} \approx 2.923$$ 6. **Find $y$ using $y = 3x - 4.5$:** $$y = 3 \times \frac{38}{13} - 4.5 = \frac{114}{13} - 4.5$$ Convert 4.5 to fraction: $$4.5 = \frac{9}{2}$$ Find common denominator 26: $$y = \frac{114}{13} - \frac{9}{2} = \frac{228}{26} - \frac{117}{26} = \frac{111}{26} \approx 4.269$$ **Final solution:** $$x = \frac{38}{13}, \quad y = \frac{111}{26}$$