1. **Stating the problem:** Simplify the expression $$\frac{0.1 + (1 - 0.1)}{1 + t \cdot 17 - 1 \cdot t \cdot \sqrt{t - 1} \cdot 19} = \frac{2}{t}$$ and verify or solve for $t$ if needed. 2. **Simplify the numerator:** $$0.1 + (1 - 0.1) = 0.1 + 0.9 = 1$$ 3. **Rewrite the denominator:** $$1 + 17t - 19t \sqrt{t - 1}$$ 4. **The equation becomes:** $$\frac{1}{1 + 17t - 19t \sqrt{t - 1}} = \frac{2}{t}$$ 5. **Cross-multiply to solve for $t$:** $$t = 2(1 + 17t - 19t \sqrt{t - 1})$$ 6. **Expand the right side:** $$t = 2 + 34t - 38t \sqrt{t - 1}$$ 7. **Bring all terms to one side:** $$t - 34t + 38t \sqrt{t - 1} = 2$$ 8. **Simplify coefficients:** $$-33t + 38t \sqrt{t - 1} = 2$$ 9. **Divide both sides by $t$ (assuming $t \neq 0$):** $$-33 + 38 \sqrt{t - 1} = \frac{2}{t}$$ 10. **Isolate the square root term:** $$38 \sqrt{t - 1} = \frac{2}{t} + 33$$ 11. **Divide both sides by 38:** $$\sqrt{t - 1} = \frac{\frac{2}{t} + 33}{38}$$ 12. **Square both sides:** $$t - 1 = \left(\frac{\frac{2}{t} + 33}{38}\right)^2$$ 13. **This is a nonlinear equation in $t$ that can be solved numerically or algebraically for $t$.** **Final simplified form:** $$\frac{0.1 + (1 - 0.1)}{1 + 17t - 19t \sqrt{t - 1}} = \frac{1}{1 + 17t - 19t \sqrt{t - 1}} = \frac{2}{t}$$ **And the key equation to solve for $t$ is:** $$t - 1 = \left(\frac{\frac{2}{t} + 33}{38}\right)^2$$