1. **State the problem:** Solve the linear equation $$-10x + 3(3x + 4) + 1 = -3(7x + 3)$$ for $x$. 2. **Apply the distributive property:** Multiply inside the parentheses. $$-10x + 3 \times 3x + 3 \times 4 + 1 = -3 \times 7x - 3 \times 3$$ which simplifies to $$-10x + 9x + 12 + 1 = -21x - 9$$ 3. **Combine like terms on the left side:** $$(-10x + 9x) + (12 + 1) = -21x - 9$$ $$-x + 13 = -21x - 9$$ 4. **Add $21x$ to both sides to get all $x$ terms on one side:** $$-x + 21x + 13 = -21x + 21x - 9$$ $$20x + 13 = -9$$ 5. **Subtract 13 from both sides:** $$20x + \cancel{13} - \cancel{13} = -9 - 13$$ $$20x = -22$$ 6. **Divide both sides by 20 to solve for $x$:** $$x = \frac{-22}{20}$$ 7. **Simplify the fraction by dividing numerator and denominator by 2:** $$x = \frac{\cancel{-22}^{11}}{\cancel{20}^{10}} = -\frac{11}{10}$$ **Final answer:** $$x = -\frac{11}{10}$$