1. **State the problem:** Solve the equation $\sin 5x - 1 = 0$ for $x$. 2. **Rewrite the equation:** Add 1 to both sides to isolate the sine term: $$\sin 5x = 1$$ 3. **Recall the sine function property:** The sine of an angle equals 1 at angles of the form: $$5x = \frac{\pi}{2} + 2k\pi, \quad k \in \mathbb{Z}$$ 4. **Solve for $x$:** Divide both sides by 5: $$x = \frac{\frac{\pi}{2} + 2k\pi}{5}$$ Show the cancellation step: $$x = \frac{\cancel{\frac{\pi}{2} + 2k\pi}}{\cancel{5}}$$ (Here, no common factors to cancel, so this is just the division step.) 5. **Final solution:** $$x = \frac{\pi}{10} + \frac{2k\pi}{5}, \quad k \in \mathbb{Z}$$ This means $x$ takes all values of $\frac{\pi}{10}$ plus integer multiples of $\frac{2\pi}{5}$. **Answer:** $$x = \frac{\pi}{10} + \frac{2k\pi}{5}, \quad k \in \mathbb{Z}$$