1. **State the problem:** Solve the linear equation $10x + 35 = 12x - 47$ for $x$. 2. **Formula and rules:** To solve for $x$, we need to isolate $x$ on one side of the equation by performing inverse operations and maintaining equality. 3. **Step 1: Move all $x$ terms to one side.** Subtract $10x$ from both sides: $$10x + 35 - 10x = 12x - 47 - 10x$$ which simplifies to $$35 = 2x - 47$$ 4. **Step 2: Move constants to the other side.** Add $47$ to both sides: $$35 + 47 = 2x - 47 + 47$$ which simplifies to $$82 = 2x$$ 5. **Step 3: Solve for $x$ by dividing both sides by 2:** $$\frac{82}{\cancel{2}} = \frac{2x}{\cancel{2}}$$ which simplifies to $$x = 41$$ 6. **Final answer:** $x = 41$