1. **State the problem:** Solve the linear equation $5x - 10 = 2x + 20$ for $x$. 2. **Write down the formula and rules:** To solve for $x$, we need to isolate $x$ on one side of the equation by performing the same operations on both sides. 3. **Step 1: Move all $x$ terms to one side.** Subtract $2x$ from both sides: $$5x - 10 - 2x = 2x + 20 - 2x$$ which simplifies to $$5x - 2x - 10 = 20$$ 4. **Step 2: Simplify the left side:** $$3x - 10 = 20$$ 5. **Step 3: Move constant terms to the other side.** Add $10$ to both sides: $$3x - 10 + 10 = 20 + 10$$ which simplifies to $$3x = 30$$ 6. **Step 4: Solve for $x$ by dividing both sides by 3:** $$\frac{3x}{\cancel{3}} = \frac{30}{\cancel{3}}$$ which simplifies to $$x = 10$$ 7. **Final answer:** $$\boxed{10}$$