1. The problem involves solving for constants $a$ and $b$ in the equations: $$2.5 = a \sqrt{b \times 25} - 5$$ and $$0 = a \sqrt{b \times (8 + 25)} - 5$$ 2. Rewrite the equations for clarity: $$2.5 = a \sqrt{25b} - 5$$ $$0 = a \sqrt{33b} - 5$$ 3. From the second equation, solve for $a$: $$0 = a \sqrt{33b} - 5 \implies a \sqrt{33b} = 5 \implies a = \frac{5}{\sqrt{33b}}$$ 4. Substitute $a$ into the first equation: $$2.5 = \frac{5}{\sqrt{33b}} \times \sqrt{25b} - 5$$ 5. Simplify the square roots: $$\sqrt{25b} = 5 \sqrt{b}$$ So, $$2.5 = \frac{5}{\sqrt{33b}} \times 5 \sqrt{b} - 5 = \frac{25 \sqrt{b}}{\sqrt{33b}} - 5$$ 6. Simplify the fraction inside the square roots: $$\frac{\sqrt{b}}{\sqrt{33b}} = \frac{1}{\sqrt{33}}$$ Therefore, $$2.5 = \frac{25}{\sqrt{33}} - 5$$ 7. Add 5 to both sides: $$2.5 + 5 = \frac{25}{\sqrt{33}}$$ $$7.5 = \frac{25}{\sqrt{33}}$$ 8. Multiply both sides by $\sqrt{33}$: $$7.5 \sqrt{33} = 25$$ 9. Solve for $\sqrt{33}$: $$\sqrt{33} = \frac{25}{7.5} = \frac{25}{7.5} = \frac{25}{7.5} = \frac{10}{3}$$ 10. But $\sqrt{33} \approx 5.7446$, which is not equal to $\frac{10}{3} \approx 3.3333$. This indicates a contradiction, so let's check the calculations again. 11. Re-examining step 6: $$\frac{\sqrt{b}}{\sqrt{33b}} = \frac{\sqrt{b}}{\sqrt{33} \sqrt{b}} = \frac{1}{\sqrt{33}}$$ This is correct. 12. So step 7 is: $$2.5 + 5 = 7.5 = \frac{25}{\sqrt{33}}$$ 13. Multiply both sides by $\sqrt{33}$: $$7.5 \sqrt{33} = 25$$ 14. Solve for $\sqrt{33}$: $$\sqrt{33} = \frac{25}{7.5} = \frac{10}{3}$$ 15. Since $\sqrt{33} \neq \frac{10}{3}$, the assumption that $a$ and $b$ are constants satisfying both equations is invalid unless $b$ is adjusted. 16. Let's solve for $b$ explicitly. From step 3: $$a = \frac{5}{\sqrt{33b}}$$ Substitute into the first equation: $$2.5 = \frac{5}{\sqrt{33b}} \times \sqrt{25b} - 5 = 5 \times \frac{\sqrt{25b}}{\sqrt{33b}} - 5 = 5 \times \sqrt{\frac{25b}{33b}} - 5 = 5 \times \sqrt{\frac{25}{33}} - 5$$ 17. Calculate $\sqrt{\frac{25}{33}}$: $$\sqrt{\frac{25}{33}} = \frac{5}{\sqrt{33}}$$ 18. So, $$2.5 = 5 \times \frac{5}{\sqrt{33}} - 5 = \frac{25}{\sqrt{33}} - 5$$ 19. Add 5 to both sides: $$7.5 = \frac{25}{\sqrt{33}}$$ 20. Multiply both sides by $\sqrt{33}$: $$7.5 \sqrt{33} = 25$$ 21. Solve for $\sqrt{33}$: $$\sqrt{33} = \frac{25}{7.5} = \frac{10}{3}$$ 22. Square both sides: $$33 = \left(\frac{10}{3}\right)^2 = \frac{100}{9}$$ 23. This is a contradiction since $33 \neq \frac{100}{9}$. Therefore, the system has no solution with the given constants. **Final conclusion:** The given equations are inconsistent for constants $a$ and $b$ as stated.