To calculate the slope (\(m\)) and intercept (\(d\)) for the line of best fit in a Lineweaver-Burk plot, we used **linear regression**, which is a statistical method to find the best-fitting straight line through a set of data points. Here’s a detailed explanation of how this was done:

### 1. **Extracting the Data Points**

The data provided was in the form of reciprocals:

\[
\text{Reciprocal Data: } \{(1/1.06, 1/3.15), (1/2.18, 1/2.66), (1/4.93, 1/8.23), (1/10.37, 1/10.59), (1/39.09, 1/15.91), (1/85.44, 1/16.16)\}
\]

First, we calculated the actual \(x\) and \(y\) values from these reciprocals:

\[
\text{Data Points: } (1.06, 3.15), (2.18, 2.66), (4.93, 8.23), (10.37, 10.59), (39.09, 15.91), (85.44, 16.16)
\]

### 2. **Applying Linear Regression**

Linear regression involves finding the slope \(m\) and y-intercept \(d\) of a line \(y = mx + d\) that minimizes the sum of squared differences between the observed \(y\) values and the values predicted by the line.

The **least squares method** was used, which computes \(m\) and \(d\) based on the following formulas:

- **Slope (\(m\))**:

\[
m = \frac{N\sum(xy) - \sum(x)\sum(y)}{N\sum(x^2) - (\sum(x))^2}
\]

- **Intercept (\(d\))**:

\[
d = \frac{\sum(y) - m\sum(x)}{N}
\]

Where:
- \(N\) is the number of data points.
- \(\sum(xy)\) is the sum of the product of \(x\) and \(y\).
- \(\sum(x)\) is the sum of all \(x\) values.
- \(\sum(y)\) is the sum of all \(y\) values.
- \(\sum(x^2)\) is the sum of the squares of \(x\) values.

### 3. **Calculations Using Python**

We used Python's `scipy.stats.linregress` function, which implements the least squares method for linear regression:

```python
from scipy.stats import linregress

# Perform linear regression to find the slope (m) and intercept (d)
slope, intercept, r_value, p_value, std_err = linregress(x_values, y_values)
```

### 4. **Results**

- **Slope (\(m\))**: \(1.4586 \times 10^{-1}\) or 0.14586
- **Intercept (\(d\))**: \(5.9719 \times 10^{0}\) or 5.9719

These values were calculated to minimize the vertical distances between the line of best fit and the actual data points.

Would you like more information on linear regression or any other topic? Here are some further questions:

1. How does the least squares method minimize error?
2. What are the assumptions behind linear regression?
3. Can linear regression be used for non-linear data?
4. How can you assess the goodness of fit in a regression model?
5. What are some limitations of using a Lineweaver-Burk plot?

**Tip:** In regression analysis, always check the residuals (differences between observed and predicted values) to ensure that they are randomly distributed and have constant variance. This helps validate the assumptions of the regression model.

Learn how to calculate the slope (m) and intercept (d) using linear regression for a Lineweaver-Burk plot. This detailed explanation includes the application of the least squares method to find the best-fitting line through a set of reciprocal data points. Suitable for advanced learners and researchers.

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