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To solve this matching problem, let's analyze each equation to understand the form of its level curves. Then, we’ll match these with the given descriptions.

1. **\( z = \sqrt{25 - x^2 - y^2} \)**  
   This is similar to the equation of a sphere or circle. Rearranged, it resembles \( x^2 + y^2 = 25 - z^2 \), which suggests circular level curves. Since \( \sqrt{25 - x^2 - y^2} \) has a maximum value, the circles will shrink as \( z \) approaches the maximum radius.  
   **Match: F. A collection of unequally spaced concentric circles**

2. **\( z = \frac{1}{x - 1} \)**  
   This equation describes a hyperbolic surface since it will diverge as \( x \) approaches 1. The level curves will consist of parallel lines spaced unequally.  
   **Match: B. A collection of unequally spaced parallel lines**

3. **\( z = xy \)**  
   This equation resembles a hyperbolic paraboloid, which results in hyperbolas for level curves. It contains both positive and negative values of \( z \) for different regions of \( x \) and \( y \).  
   **Match: D. Two straight lines and a collection of hyperbolas**

4. **\( z = 2x^2 + 3y^2 \)**  
   This resembles an elliptical paraboloid, where the level curves will be ellipses due to the different coefficients of \( x^2 \) and \( y^2 \).  
   **Match: A. A collection of concentric ellipses**

5. **\( z = x^2 + y^2 \)**  
   This is a standard equation for a paraboloid with circular level curves. The level curves will be equally spaced concentric circles as \( z \) increases.  
   **Match: C. A collection of equally spaced concentric circles**

6. **\( z = 2x + 3y \)**  
   This is a plane equation, where level curves are straight lines. Since the coefficients are constant, the lines will be equally spaced.  
   **Match: E. A collection of equally spaced parallel lines**

7. **\( z = \sqrt{x^2 + y^2} \)**  
   This equation represents a cone, where the level curves are circles. Since it's a square root, they will be unequally spaced.  
   **Match: F. A collection of unequally spaced concentric circles**

### Final Answers:
1. **F**
2. **B**
3. **D**
4. **A**
5. **C**
6. **E**
7. **F**

Would you like a further breakdown of any specific equation or concept?

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Here are some related questions to deepen understanding:

1. How do we determine the spacing of level curves based on the function's form?
2. What are the differences between equally and unequally spaced concentric circles in level curves?
3. How does an elliptical paraboloid differ from a hyperbolic paraboloid in terms of level curves?
4. What does it mean for level curves to be "concentric"?
5. How can you identify a hyperbolic surface from its equation?

**Tip:** Remember that the presence of square terms \( x^2 \) and \( y^2 \) often indicates circular or elliptical level curves, while linear terms often indicate straight lines.

Match the surfaces with the verbal description of the level curves by placing the letter of the verbal description to the left of the number of the surface.

Learn how to match different surface equations with their corresponding level curve descriptions. This problem involves understanding level curves for various surfaces, such as concentric circles, ellipses, parallel lines, and hyperbolas.

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