1. **State the problem:** Solve the linear equation $6x - 2 = x + 3$ for $x$. 2. **Write down the formula and rules:** To solve for $x$, we want to isolate $x$ on one side of the equation. We can do this by moving all terms involving $x$ to one side and constants to the other. 3. **Step 1: Subtract $x$ from both sides to get all $x$ terms on the left:** $$6x - 2 - x = x + 3 - x$$ which simplifies to $$5x - 2 = 3$$ 4. **Step 2: Add 2 to both sides to move constants to the right:** $$5x - 2 + 2 = 3 + 2$$ which simplifies to $$5x = 5$$ 5. **Step 3: Divide both sides by 5 to solve for $x$:** $$\frac{5x}{\cancel{5}} = \frac{5}{\cancel{5}}$$ which simplifies to $$x = 1$$ 6. **Final answer:** $$\boxed{1}$$