1. **State the problem:** Solve the linear equation $$-5x + 7 = -1$$ for $x$. 2. **Isolate the variable term:** Subtract 7 from both sides to get $$-5x = -1 - 7$$. 3. **Simplify the right side:** $$-5x = -8$$. 4. **Solve for $x$:** Divide both sides by $-5$ to isolate $x$: $$x = \frac{-8}{-5}$$. 5. **Simplify the fraction:** $$x = \frac{8}{5}$$. **Final answer:** $$x = \frac{8}{5}$$.