The slope of the function \( y = f(x, z) = 25 - 0.5x + 3z \) can be interpreted in the context of partial derivatives, as this function depends on two variables, \( x \) and \( z \).

1. **Partial derivative with respect to \( x \)**:
   \[
   \frac{\partial y}{\partial x} = \frac{\partial}{\partial x} \left( 25 - 0.5x + 3z \right) = -0.5
   \]
   This result means that for each unit increase in \( x \), the value of \( y \) decreases by 0.5, assuming \( z \) remains constant.

2. **Partial derivative with respect to \( z \)**:
   \[
   \frac{\partial y}{\partial z} = \frac{\partial}{\partial z} \left( 25 - 0.5x + 3z \right) = 3
   \]
   This result means that for each unit increase in \( z \), the value of \( y \) increases by 3, assuming \( x \) remains constant.

### Summary:
- The slope with respect to \( x \) is \(-0.5\).
- The slope with respect to \( z \) is \( 3 \).

Would you like more details on partial derivatives or how to interpret these slopes?

Here are some related questions you might consider:
1. What is the geometric interpretation of partial derivatives?
2. How do you find the gradient of a multivariable function?
3. What does the gradient tell you about the direction of steepest ascent?
4. How do you determine the slope of a function along a specific direction?
5. How do you find the second partial derivatives, and what do they represent?

**Tip:** Understanding partial derivatives is crucial for analyzing how changes in one variable affect the function while keeping the other variables constant.

Explore how to interpret the slope of the function y = 25 - 0.5x + 3z through partial derivatives. Learn the impacts of changes in x and z on the function's value. Suitable for advanced college students and anyone interested in multivariable calculus.

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