1. Stating the problem: We are given the system of equations $$5x + y = 13$$ and $$5xy = 13$$, and asked to find the value of $$12.5x^2 + y^2$$ given that $$xy = 10$$. 2. Step 1: Analyze the given equations. We have: $$5x + y = 13$$ $$5xy = 13$$ and $$xy = 10$$. 3. Step 2: Check compatibility. From $$5xy = 13$$, substituting $$xy = 10$$, $$5 \times 10 = 50 \neq 13$$. This means the condition $$5xy = 13$$ contradicts $$xy = 10$$. Both cannot be true simultaneously. 4. Step 3: Adjust understanding. Since $$xy = 10$$ is required, but $$5xy=13$$ conflicts, we focus only on $$xy = 10$$ and $$5x + y = 13$$. 5. Step 4: Use variables to solve for y. From $$5x + y = 13$$, we get: $$y = 13 - 5x$$. 6. Step 5: Substitute $$y$$ into $$xy = 10$$: $$x(13 - 5x) = 10$$ $$13x - 5x^2 = 10$$ $$5x^2 - 13x + 10 = 0$$. 7. Step 6: Solve the quadratic equation. The quadratic is: $$5x^2 - 13x + 10 = 0$$ Using the quadratic formula: $$x = \frac{13 \pm \sqrt{(-13)^2 - 4 \times 5 \times 10}}{2 \times 5} = \frac{13 \pm \sqrt{169 - 200}}{10} = \frac{13 \pm \sqrt{-31}}{10}$$ Since the discriminant is negative, the solutions for $$x$$ are complex, meaning no real solution exists that satisfies both conditions. 8. Step 7: Conclusion. No real $$x,y$$ satisfy both $$5x + y = 13$$ and $$xy = 10$$ simultaneously, so the value of $$12.5x^2 + y^2$$ cannot be found under these constraints. Final answer: No real solution exists given the problem's constraints, so $$12.5x^2 + y^2$$ cannot be determined.