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The problem appears to involve fitting a straight line to a set of data points for coal production across different years. This uses the **method of least squares** to find a line of best fit for the data. Here’s the step-by-step solution.

### Step 1: Define Variables
The goal is to fit the line \( y = a + bx \), where:
- \( y \) is the production in thousand tonnes,
- \( x \) is the year transformed to simplify the calculations (often centered around a base year).

From the image:
- **Origin year is 1983**. We define \( x = 0 \) at 1983. For each year:
  - \( x = -4 \) for 1979,
  - \( x = -2 \) for 1981,
  - \( x = -1 \) for 1982,
  - \( x = 1 \) for 1984,
  - \( x = 2 \) for 1985, and so on.

### Step 2: Summarize the Data
Using the provided table:
- **Sum of \( x \) values** (which should be zero since we chose a central origin),
- **Sum of \( y \) values**: The total production values,
- **Sum of \( x^2 \)**: Sum of squared \( x \) values,
- **Sum of \( xy \)**: Sum of the product of each \( x \) and \( y \) value.

### Step 3: Apply the Least Squares Formulas
The formulas to determine the coefficients \( a \) and \( b \) in \( y = a + bx \) are:
\[
b = \frac{\sum xy}{\sum x^2}
\]
\[
a = \frac{\sum y}{n}
\]
where \( n \) is the number of data points.

From the table:
- The values for \( \sum y \), \( \sum x^2 \), and \( \sum xy \) are likely calculated already, and we can substitute them in to find \( a \) and \( b \).

### Step 4: Interpret the Trend Line
With \( a \) and \( b \) determined, we have the trend line equation that models the production of coal over time. The slope \( b \) represents the yearly increase in production, and to find the **monthly increase**, divide \( b \) by 12.

Would you like me to go through each calculation in detail, or do you have specific questions about any part of this process? 

Here are some questions that could further expand this topic:
1. How do we interpret the slope \( b \) in the context of this problem?
2. What are alternative methods for trend analysis besides least squares?
3. Why is 1983 chosen as the origin, and how does it affect the calculations?
4. How can we predict future production values using this trend line?
5. How does dividing \( b \) by 12 yield the monthly increase?

**Tip**: Centering \( x \) around a midpoint (like 1983 here) often simplifies calculations by making \( \sum x = 0 \).

Fit a straight line by the method of least squares and calculate the trend values for the given production data.

Learn how to fit a straight line to coal production data using the least squares method. This tutorial covers linear regression techniques, calculation of trend values, and interpretation of the slope for yearly and monthly increases in production.

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