1. **State the problem:** We are given two equations: $$0 = 7.6 \left( \frac{1}{1 + \frac{R_u}{1000}} - \frac{1}{1+r} \right) \quad \text{(Equation ①)}$$ $$0.5 = 7.6 \left( \frac{1}{1 + \frac{R_u}{1300}} - \frac{1}{1+r} \right) \quad \text{(Equation ②)}$$ We need to find the values of $R_u$ and $r$ that satisfy these equations. 2. **Rewrite the equations for clarity:** Let’s denote: $$A = \frac{1}{1 + \frac{R_u}{1000}} = \frac{1000}{1000 + R_u}$$ $$B = \frac{1}{1 + r}$$ $$C = \frac{1}{1 + \frac{R_u}{1300}} = \frac{1300}{1300 + R_u}$$ Then the equations become: $$0 = 7.6 (A - B) \implies A = B$$ $$0.5 = 7.6 (C - B)$$ 3. **From Equation ①:** $$A = B \implies \frac{1000}{1000 + R_u} = \frac{1}{1 + r}$$ Cross-multiplied: $$1000 (1 + r) = 1000 + R_u$$ $$1000 + 1000r = 1000 + R_u$$ Simplify: $$1000r = R_u \implies r = \frac{R_u}{1000}$$ 4. **Substitute $r = \frac{R_u}{1000}$ into Equation ②:** $$0.5 = 7.6 \left( \frac{1300}{1300 + R_u} - \frac{1}{1 + \frac{R_u}{1000}} \right)$$ Recall: $$\frac{1}{1 + \frac{R_u}{1000}} = \frac{1000}{1000 + R_u}$$ So: $$0.5 = 7.6 \left( \frac{1300}{1300 + R_u} - \frac{1000}{1000 + R_u} \right)$$ 5. **Divide both sides by 7.6:** $$\frac{0.5}{7.6} = \frac{1300}{1300 + R_u} - \frac{1000}{1000 + R_u}$$ Calculate left side: $$\approx 0.06579$$ 6. **Set up the equation:** $$0.06579 = \frac{1300}{1300 + R_u} - \frac{1000}{1000 + R_u}$$ 7. **Find common denominator and simplify:** Multiply both sides by $(1300 + R_u)(1000 + R_u)$: $$0.06579 (1300 + R_u)(1000 + R_u) = 1300(1000 + R_u) - 1000(1300 + R_u)$$ 8. **Simplify right side:** $$1300(1000 + R_u) - 1000(1300 + R_u) = 1,300,000 + 1300 R_u - 1,300,000 - 1000 R_u = 300 R_u$$ 9. **Expand left side:** $$(1300 + R_u)(1000 + R_u) = 1,300,000 + 2300 R_u + R_u^2$$ So: $$0.06579 (1,300,000 + 2300 R_u + R_u^2) = 300 R_u$$ 10. **Distribute:** $$85,527 + 151.317 R_u + 0.06579 R_u^2 = 300 R_u$$ 11. **Bring all terms to one side:** $$0.06579 R_u^2 + 151.317 R_u - 300 R_u + 85,527 = 0$$ $$0.06579 R_u^2 - 148.683 R_u + 85,527 = 0$$ 12. **Solve quadratic equation:** Use quadratic formula: $$R_u = \frac{148.683 \pm \sqrt{(-148.683)^2 - 4 \times 0.06579 \times 85,527}}{2 \times 0.06579}$$ Calculate discriminant: $$= 22111.5 - 22512.4 = -400.9$$ Since discriminant is negative, no real solution exists for $R_u$. 13. **Interpretation:** No real $R_u$ satisfies both equations simultaneously under these assumptions. **Final answer:** There is no real solution for $R_u$ and $r$ that satisfies both equations simultaneously. **Slug:** solve Ru r **Subject:** algebra