https://file.aiquickdraw.com/mathbot/file/172999242728698sxth81.jpg

To solve this problem, let's analyze the given system of equations and the statements provided. Here’s a breakdown of the key points and the solution.

### Problem Analysis:
1. **Parabola Equation**: \( y = -(x - 4)^2 + 9 \)
   - This is a downward-facing parabola with its vertex at \( (4, 9) \).
   
2. **Line Equation**: The yellow line is not explicitly given an equation in the image, but we can infer information based on the intersections marked.

### Solution Analysis:
The question is asking for the points where the line intersects the parabola. Solutions to the system of equations are where both equations have the same \( x \) and \( y \) values.

### Evaluate Each Statement:
1. **First Statement**: A solution exists at \( (0, 9) \) where the line crosses the y-axis.
   - Check if \( (0, 9) \) lies on the parabola by substituting \( x = 0 \) into the parabola's equation:
     \[
     y = -(0 - 4)^2 + 9 = -16 + 9 = -7
     \]
   - Since \( (0, 9) \) is not on the parabola, this statement is incorrect.

2. **Second Statement**: A solution exists at \( (4.5, 0) \) where the line crosses the x-axis.
   - Check if \( (4.5, 0) \) is on the parabola by substituting \( x = 4.5 \) into the equation:
     \[
     y = -(4.5 - 4)^2 + 9 = -0.25 + 9 = 8.75
     \]
   - Since \( (4.5, 0) \) is not on the parabola, this statement is incorrect.

3. **Third Statement**: A solution exists at \( (2, 5) \) where the two graphs intersect.
   - Substitute \( x = 2 \) in the parabola's equation to check if \( y = 5 \):
     \[
     y = -(2 - 4)^2 + 9 = -4 + 9 = 5
     \]
   - Since \( (2, 5) \) lies on the parabola, this point is indeed an intersection of the line and the parabola, so this statement is **correct**.

4. **Fourth Statement**: A solution exists at point \( L \) where the parabola crosses the y-axis.
   - The parabola crosses the y-axis when \( x = 0 \), and as we calculated earlier, the y-coordinate is \( -7 \), not matching any line intersection. This statement is incorrect.

5. **Fifth Statement**: A solution exists at point \( N \) where the parabola crosses the x-axis.
   - To find where the parabola crosses the x-axis, set \( y = 0 \):
     \[
     0 = -(x - 4)^2 + 9
     \]
     \[
     (x - 4)^2 = 9
     \]
     \[
     x - 4 = \pm 3 \Rightarrow x = 7 \text{ or } x = 1
     \]
   - The points \( (1, 0) \) and \( (7, 0) \) are the x-axis intersections of the parabola, not on point \( N \) marked on the line. This statement is incorrect.

6. **Sixth Statement**: A solution exists at \( (8, -6) \) where the two graphs intersect.
   - Substitute \( x = 8 \) in the parabola's equation:
     \[
     y = -(8 - 4)^2 + 9 = -16 + 9 = -7
     \]
   - This is close to \( -6 \), but does not exactly match. This statement is incorrect.

### Conclusion:
The correct statement is:
- **A solution exists at (2, 5) where the two graphs intersect.**

Would you like further details or have any questions?

### Additional Questions:
1. What would be the process for finding the exact equation of the yellow line in this system?
2. How can we algebraically confirm if other intersections exist between the line and the parabola?
3. How does shifting the vertex of a parabola affect its intersections with a line?
4. What are the general steps to solve a system of equations involving a parabola and a line?
5. How do we interpret the discriminant of the quadratic equation derived from the intersection points?

**Tip**: When solving systems involving parabolas and lines, look for symmetry and key points like vertices and axis intersections to simplify calculations.

Select the correct statement in the table. Consider the graph of a system of equations, where the parabola models the equation y = -(x - 4)^2 + 9. Determine the true statement.

Analyze a system of equations to find intersection points between a parabola modeled by y = -(x - 4)^2 + 9 and a line. This problem helps students understand how to determine where a parabola and a line intersect and evaluate solution points for accuracy.

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