1. **State the problem:** Solve the linear equation $$3(v + 4) = -3(4v - 8) + 5v$$ for the variable $v$. 2. **Apply the distributive property:** Multiply each term inside the parentheses by the factor outside. $$3 \times v + 3 \times 4 = -3 \times 4v + (-3) \times (-8) + 5v$$ which simplifies to $$3v + 12 = -12v + 24 + 5v$$ 3. **Combine like terms on the right side:** $$3v + 12 = (-12v + 5v) + 24$$ $$3v + 12 = -7v + 24$$ 4. **Add $7v$ to both sides to get all $v$ terms on one side:** $$3v + 7v + 12 = -7v + 7v + 24$$ $$10v + 12 = 24$$ 5. **Subtract 12 from both sides to isolate the term with $v$:** $$10v + \cancel{12} - \cancel{12} = 24 - 12$$ $$10v = 12$$ 6. **Divide both sides by 10 to solve for $v$:** $$\frac{10v}{\cancel{10}} = \frac{12}{10}$$ $$v = \frac{12}{10}$$ 7. **Simplify the fraction:** $$v = \frac{6}{5}$$ **Final answer:** $$v = \frac{6}{5}$$