1. **Problem statement:** Solve the system of equations: $$x^2 - y^2 = \frac{15\sqrt{x} - 17\sqrt{y}}{4\sqrt{xy}}$$ $$x^2 + 14xy + y^2 = \frac{17\sqrt{x} + 15\sqrt{y}}{x+y}$$ 2. **Introduce substitutions:** Let $a = \sqrt{x}$ and $b = \sqrt{y}$, so $x = a^2$ and $y = b^2$. 3. **Rewrite the equations in terms of $a$ and $b$:** - First equation: $$x^2 - y^2 = a^4 - b^4 = (a^2 - b^2)(a^2 + b^2)$$ - The right side: $$\frac{15a - 17b}{4ab}$$ So the first equation becomes: $$ (a^2 - b^2)(a^2 + b^2) = \frac{15a - 17b}{4ab} $$ 4. **Rewrite the second equation:** $$x^2 + 14xy + y^2 = a^4 + 14a^2b^2 + b^4 = (a^2 + b^2)^2 + 12a^2b^2$$ The right side: $$\frac{17a + 15b}{a^2 + b^2}$$ So the second equation becomes: $$ (a^2 + b^2)^2 + 12a^2b^2 = \frac{17a + 15b}{a^2 + b^2} $$ 5. **Simplify the first equation:** Multiply both sides by $4ab$: $$4ab(a^2 - b^2)(a^2 + b^2) = 15a - 17b$$ 6. **Simplify the second equation:** Multiply both sides by $a^2 + b^2$: $$ (a^2 + b^2)^3 + 12a^2b^2(a^2 + b^2) = 17a + 15b $$ 7. **Try to find solutions by inspection:** Assume $a$ and $b$ are positive real numbers. Try $a = b$: - Then $a^2 - b^2 = 0$, so left side of first equation is zero. - Right side of first equation: $$\frac{15a - 17a}{4a^2} = \frac{-2a}{4a^2} = -\frac{1}{2a} \neq 0$$ So $a \neq b$. 8. **Try $a = 1$, solve for $b$ from first equation:** $$4(1)(b)(1 - b^2)(1 + b^2) = 15(1) - 17b$$ $$4b(1 - b^2)(1 + b^2) = 15 - 17b$$ Note that $(1 - b^2)(1 + b^2) = 1 - b^4$. So: $$4b(1 - b^4) = 15 - 17b$$ $$4b - 4b^5 = 15 - 17b$$ Bring all terms to one side: $$4b - 4b^5 + 17b - 15 = 0$$ $$21b - 4b^5 - 15 = 0$$ 9. **Try $b = 1$:** $$21(1) - 4(1) - 15 = 21 - 4 - 15 = 2 \neq 0$$ Try $b = \frac{3}{2} = 1.5$: $$21(1.5) - 4(1.5)^5 - 15$$ Calculate $1.5^5 = 7.59375$: $$31.5 - 4(7.59375) - 15 = 31.5 - 30.375 - 15 = -13.875 \neq 0$$ Try $b = 1.2$: $$21(1.2) - 4(1.2)^5 - 15$$ Calculate $1.2^5 = 2.48832$: $$25.2 - 9.95328 - 15 = 0.24672 \approx 0$$ Close to zero, so $b \approx 1.2$. 10. **Check second equation with $a=1$, $b=1.2$:** Calculate left side: $$(1 + 1.44)^3 + 12(1)(1.44)(1 + 1.44) = (2.44)^3 + 12(1.44)(2.44)$$ $$= 14.525 + 42.163 = 56.688$$ Calculate right side: $$17(1) + 15(1.2) = 17 + 18 = 35$$ Not equal, so $a=1$, $b=1.2$ is not a solution. 11. **Try $b = 1$, solve for $a$ from first equation:** $$4a( a^2 - 1)( a^2 + 1) = 15a - 17$$ $$(a^2 - 1)(a^2 + 1) = a^4 - 1$$ So: $$4a(a^4 - 1) = 15a - 17$$ $$4a^5 - 4a = 15a - 17$$ $$4a^5 - 19a + 17 = 0$$ Try $a=1$: $$4 - 19 + 17 = 2 \neq 0$$ Try $a=2$: $$4(32) - 38 + 17 = 128 - 38 + 17 = 107 \neq 0$$ Try $a=1.5$: $$4(7.59375) - 28.5 + 17 = 30.375 - 28.5 + 17 = 18.875 \neq 0$$ Try $a=1.2$: $$4(2.48832) - 22.8 + 17 = 9.953 - 22.8 + 17 = 4.153 \neq 0$$ 12. **Try to find rational solutions by guessing:** Try $a=\frac{17}{15}$ and $b=1$: Calculate left side of first equation: $$4 \times \frac{17}{15} \times \left(\left(\frac{17}{15}\right)^2 - 1\right) \times \left(\left(\frac{17}{15}\right)^2 + 1\right)$$ Calculate $\left(\frac{17}{15}\right)^2 = \frac{289}{225}$ So: $$4 \times \frac{17}{15} \times \left(\frac{289}{225} - 1\right) \times \left(\frac{289}{225} + 1\right) = 4 \times \frac{17}{15} \times \frac{64}{225} \times \frac{514}{225}$$ Calculate numerator: $$4 \times 17 \times 64 \times 514 = 4 \times 17 \times 32896 = 4 \times 559232 = 2236928$$ Denominator: $$15 \times 225 \times 225 = 15 \times 50625 = 759375$$ So left side: $$\frac{2236928}{759375} \approx 2.947$$ Right side: $$15 \times \frac{17}{15} - 17 \times 1 = 17 - 17 = 0$$ Not equal, so no. 13. **Summary:** The system is complicated and likely requires numerical or advanced algebraic methods. 14. **Final answer:** The system has no simple closed-form solution with elementary methods. Numerical approximation or computer algebra system is recommended. **Note:** The problem is complex and may require iterative or numerical methods beyond this scope.