1. **Problem statement:** Solve the equation $12x - 3 = 8x + 1$ for $x$. 2. **Formula and rules:** To solve linear equations, we isolate the variable $x$ by performing the same operations on both sides of the equation. 3. **Step 1:** Subtract $8x$ from both sides: $$12x - 3 - 8x = 8x + 1 - 8x$$ which simplifies to $$12x - \cancel{8x} - 3 = \cancel{8x} + 1$$ $$4x - 3 = 1$$ 4. **Step 2:** Add 3 to both sides: $$4x - 3 + 3 = 1 + 3$$ $$4x = 4$$ 5. **Step 3:** Divide both sides by 4: $$\frac{4x}{\cancel{4}} = \frac{4}{\cancel{4}}$$ $$x = 1$$ **Final answer:** $$x = 1$$