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The given equation is:

\[
y = -Ae^{-Cx} + B
\]

To linearize this equation, we aim to express it in a form that can be plotted as a straight line, such as \( Y = mX + c \), where \(m\) is the slope and \(c\) is the intercept.

### Steps to linearize:
1. **Isolate the exponential term**:
   Subtract \( B \) from both sides:

   \[
   y - B = -Ae^{-Cx}
   \]

2. **Eliminate the negative sign and divide by \( A \)**:

   \[
   \frac{B - y}{A} = e^{-Cx}
   \]

3. **Apply the natural logarithm (\(\ln\)) to both sides** to remove the exponential:

   \[
   \ln\left(\frac{B - y}{A}\right) = -Cx
   \]

This is now a linear form with respect to \(x\), where:

- \( Y = \ln\left(\frac{B - y}{A}\right) \)
- \( m = -C \) (the slope)
- \( X = x \)

Thus, the equation is linearized as:

\[
\ln\left(\frac{B - y}{A}\right) = -Cx
\]

This equation can be plotted with \( x \) on the horizontal axis and \( \ln\left(\frac{B - y}{A}\right) \) on the vertical axis. The slope of this line will give you \( -C \), and the intercept can provide additional insight depending on the problem.

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Would you like further details or have any specific questions about this?

Here are some related questions you could explore:
1. How does the slope \( C \) affect the rate of decay in the original equation?
2. What does the intercept of the linearized form represent physically?
3. How can this form help in fitting data to an exponential model?
4. How would you find the constants \( A \), \( B \), and \( C \) from experimental data?
5. What are the practical applications of such exponential decay equations?

**Tip**: Linearizing exponential functions is a common technique in data analysis, making it easier to determine parameters through simple linear regression methods.

linearise this equation: y = -Ae^(-Cx) + B

Learn how to linearize the exponential equation y = -Ae^(-Cx) + B. This problem involves applying logarithmic transformations to express the equation in a linear form suitable for regression analysis, covering exponential functions and logarithmic properties.

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