1. **State the problem:** Solve the equation $10^{-3x+1} = \frac{1}{10}$. 2. **Recall the properties of exponents:** We know that $\frac{1}{10} = 10^{-1}$ because $10^{-1} = \frac{1}{10}$. 3. **Rewrite the equation using the same base:** $$10^{-3x+1} = 10^{-1}$$ 4. **Set the exponents equal:** Since the bases are the same and nonzero, the exponents must be equal: $$-3x + 1 = -1$$ 5. **Solve for $x$:** $$-3x + 1 = -1$$ $$-3x = -1 - 1$$ $$-3x = -2$$ 6. **Divide both sides by $-3$:** $$x = \frac{-2}{\cancel{-3}} \quad \Rightarrow \quad x = \frac{\cancel{-2}}{3} \quad \Rightarrow \quad x = \frac{2}{3}$$ **Final answer:** $$x = \frac{2}{3}$$