1. **State the problem:** Solve the equation $x - 3 = 5x + 7$ for $x$. 2. **Write down the equation:** $$x - 3 = 5x + 7$$ 3. **Goal:** Isolate $x$ on one side of the equation. 4. **Subtract $x$ from both sides to get all $x$ terms on the right:** $$x - 3 - x = 5x + 7 - x$$ which simplifies to $$-3 = 4x + 7$$ 5. **Subtract 7 from both sides to isolate the term with $x$:** $$-3 - 7 = 4x + 7 - 7$$ which simplifies to $$-10 = 4x$$ 6. **Divide both sides by 4 to solve for $x$:** $$x = \frac{-10}{4}$$ 7. **Simplify the fraction:** $$x = -\frac{5}{2}$$ **Final answer:** $$x = -\frac{5}{2}$$