1. **State the problem:** Solve the linear equation $$-8x + 1 = 17$$ for $x$. 2. **Isolate the variable term:** Subtract 1 from both sides to get rid of the constant on the left. $$-8x + 1 - 1 = 17 - 1$$ $$-8x = 16$$ 3. **Divide both sides by the coefficient of $x$:** The coefficient is $-8$, so divide both sides by $-8$. $$x = \frac{16}{-8}$$ 4. **Simplify the fraction:** $$x = \frac{\cancel{16}^{2 \times 8}}{\cancel{-8}^{-1 \times 8}} = -2$$ 5. **Final answer:** $$x = -2$$ This means the value of $x$ that satisfies the equation is $-2$.