1. **State the problem:** Solve the equation $$\sqrt{5x-34} - 17 = -16$$. 2. **Isolate the square root term:** Add 17 to both sides to isolate the square root. $$\sqrt{5x-34} - 17 + 17 = -16 + 17$$ $$\sqrt{5x-34} = 1$$ 3. **Square both sides:** To eliminate the square root, square both sides of the equation. $$\left(\sqrt{5x-34}\right)^2 = 1^2$$ $$5x - 34 = 1$$ 4. **Solve for $x$:** Add 34 to both sides. $$5x - 34 + 34 = 1 + 34$$ $$5x = 35$$ Divide both sides by 5. $$\frac{5x}{\cancel{5}} = \frac{35}{\cancel{5}}$$ $$x = 7$$ 5. **Check the solution:** Substitute $x=7$ back into the original equation. $$\sqrt{5(7) - 34} - 17 = \sqrt{35 - 34} - 17 = \sqrt{1} - 17 = 1 - 17 = -16$$ The left side equals the right side, so $x=7$ is the correct solution. **Final answer:** $$x = 7$$