P=(1-K/D)f

what kind of mathematical transformations can be applied to the above. show me all

The expression \( P = \left(1 - \frac{K}{D}\right)f \) can undergo various mathematical transformations depending on the purpose, such as simplification, expansion, factorization, or solving for a particular variable. Here are the main transformations that can be applied:

### 1. **Expanding the Expression**:
   You can distribute the multiplication over subtraction:
   \[
   P = f \cdot 1 - f \cdot \frac{K}{D} = f - \frac{fK}{D}
   \]
   The expanded form is:
   \[
   P = f - \frac{fK}{D}
   \]

### 2. **Factoring the Expression**:
   The original form is already factored with respect to \( f \):
   \[
   P = f\left(1 - \frac{K}{D}\right)
   \]
   Alternatively, you can factor out \( -f \):
   \[
   P = -f\left(\frac{K}{D} - 1\right)
   \]

### 3. **Solving for \( f \)**:
   To solve for \( f \), you would isolate it:
   \[
   f = \frac{P}{1 - \frac{K}{D}} 
   \]
   Simplify further:
   \[
   f = \frac{PD}{D - K}
   \]

### 4. **Solving for \( K \)**:
   To solve for \( K \), first isolate \( \frac{K}{D} \):
   \[
   \frac{K}{D} = 1 - \frac{P}{f}
   \]
   Multiply both sides by \( D \):
   \[
   K = D\left(1 - \frac{P}{f}\right) = D - \frac{PD}{f}
   \]

### 5. **Solving for \( D \)**:
   To solve for \( D \), isolate it:
   \[
   1 - \frac{K}{D} = \frac{P}{f}
   \]
   Simplify and rearrange:
   \[
   \frac{K}{D} = 1 - \frac{P}{f}
   \]
   Invert and solve:
   \[
   D = \frac{Kf}{f - P}
   \]

### Tip:
Always check the domain of the variables, especially when dividing by expressions that might equal zero.

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Do you want further details or have any specific questions about these transformations? Here are some additional related questions you might consider:

1. How can the expression be transformed using logarithms if possible?
2. What happens if we substitute specific values for \( K \), \( D \), and \( f \)?
3. How can we interpret this expression graphically with respect to one variable?
4. How would the transformations change if the expression were part of a larger equation?
5. What are the conditions for \( P \) to be zero?



Explore mathematical transformations of the expression P=(1-K/D)f. Understand how to expand, factor, and solve for variables in this equation. Suitable for advanced high school students and above.

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