1. We start with the first equation: $x - \sqrt{x^2 - 5} = 1$. 2. Isolate the square root term: $\sqrt{x^2 - 5} = x - 1$. 3. Note that the expression under the square root must be non-negative and $x - 1 \geq 0$ for the right side to be valid, so $x \geq 1$. 4. Square both sides to eliminate the square root: $$\left(\sqrt{x^2 - 5}\right)^2 = (x - 1)^2$$ $$x^2 - 5 = (x - 1)^2$$ 5. Expand the right side: $$x^2 - 5 = x^2 - 2x + 1$$ 6. Subtract $x^2$ from both sides: $$\cancel{x^2} - 5 = \cancel{x^2} - 2x + 1$$ $$-5 = -2x + 1$$ 7. Add $2x$ to both sides and add 5 to both sides: $$2x = 1 + 5$$ $$2x = 6$$ 8. Divide both sides by 2: $$x = \frac{6}{2}$$ $$x = 3$$ 9. Check the solution in the original equation to avoid extraneous roots: $$3 - \sqrt{3^2 - 5} = 3 - \sqrt{9 - 5} = 3 - \sqrt{4} = 3 - 2 = 1$$ This satisfies the equation. Final answer: $x = 3$