1. **State the problem:** Solve the system of equations: $$x + y = 5$$ $$xy = 25$$ 2. **Use the formulas:** We want to find values of $x$ and $y$ that satisfy both equations simultaneously. 3. **Express $y$ from the first equation:** $$y = 5 - x$$ 4. **Substitute $y$ into the second equation:** $$x(5 - x) = 25$$ 5. **Expand and simplify:** $$5x - x^2 = 25$$ 6. **Rewrite as a quadratic equation:** $$-x^2 + 5x - 25 = 0$$ 7. **Multiply both sides by $-1$ to simplify:** $$\cancel{-}x^2 + \cancel{5}x - \cancel{25} = \cancel{0}$$ becomes $$x^2 - 5x + 25 = 0$$ 8. **Calculate the discriminant $\Delta$:** $$\Delta = b^2 - 4ac = (-5)^2 - 4 \times 1 \times 25 = 25 - 100 = -75$$ 9. **Interpret the discriminant:** Since $\Delta < 0$, there are no real solutions for $x$ and $y$. 10. **Conclusion:** The system has no real solutions because the product $xy=25$ and sum $x+y=5$ cannot be satisfied simultaneously with real numbers.