1. Stating the problem: Solve the linear equation $3x - 4 \frac{1}{2} = -19 \frac{1}{2}$. 2. Convert mixed numbers to improper fractions or decimals for easier calculation: $4 \frac{1}{2} = 4.5$ and $-19 \frac{1}{2} = -19.5$. 3. Rewrite the equation: $$3x - 4.5 = -19.5$$ 4. Isolate the term with $x$ by adding $4.5$ to both sides: $$3x = -19.5 + 4.5$$ $$3x = -15$$ 5. Solve for $x$ by dividing both sides by $3$: $$x = \frac{-15}{3}$$ $$x = -5$$ 6. Final answer: $x = -5$.