1. **State the problem:** Solve the linear equation $3x - 6 = 12x + 9$ for $x$. 2. **Formula and rules:** To solve linear equations, we isolate the variable on one side by performing inverse operations such as addition, subtraction, multiplication, or division. 3. **Step-by-step solution:** - Start with the equation: $$3x - 6 = 12x + 9$$ - Subtract $3x$ from both sides to get all $x$ terms on one side: $$3x - 6 - 3x = 12x + 9 - 3x$$ $$-6 = 9x + 9$$ - Subtract $9$ from both sides to isolate the term with $x$: $$-6 - 9 = 9x + 9 - 9$$ $$-15 = 9x$$ - Divide both sides by $9$ to solve for $x$: $$x = \frac{-15}{9}$$ - Simplify the fraction: $$x = \frac{-5}{3}$$ 4. **Final answer:** $$x = -\frac{5}{3}$$ This means the value of $x$ that satisfies the equation is $-\frac{5}{3}$.