1. **State the problem:** Imani answered a total of 27 questions correctly in two categories: standard and challenge. Each standard question is worth 15 points, each challenge question is worth 50 points, and she earned a total of 580 points. 2. **Define variables:** Let $x$ be the number of standard questions answered correctly, and $y$ be the number of challenge questions answered correctly. 3. **Write the system of equations:** $$\begin{cases} x + y = 27 \\ 15x + 50y = 580 \end{cases}$$ 4. **Solve the first equation for $x$:** $$x = 27 - y$$ 5. **Substitute $x$ into the second equation:** $$15(27 - y) + 50y = 580$$ 6. **Distribute and simplify:** $$405 - 15y + 50y = 580$$ 7. **Combine like terms:** $$405 + 35y = 580$$ 8. **Isolate $y$:** $$35y = 580 - 405$$ $$35y = 175$$ 9. **Divide both sides by 35:** $$y = \frac{\cancel{35}175}{\cancel{35}1} = 5$$ 10. **Find $x$ using $x = 27 - y$:** $$x = 27 - 5 = 22$$ 11. **Answer:** Imani answered **22** standard questions correctly and **5** challenge questions correctly.