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Let's break this down step by step:

1. **Determine the constant rate of change** (part a):

   The rate of change (also known as the slope) is given by the formula:
   \[
   \frac{\Delta k}{\Delta j} = \frac{k_2 - k_1}{j_2 - j_1}
   \]
   where \( \Delta k \) is the change in \( k \) values, and \( \Delta j \) is the change in \( j \) values.

   Using the given values from the table:
   - For \( j = 6 \) and \( k = 0.2 \)
   - For \( j = -2 \) and \( k = 18.6 \)

   Let's calculate the slope:
   \[
   \frac{\Delta k}{\Delta j} = \frac{0.2 - 18.6}{6 - (-2)} = \frac{-18.4}{8} = -2.3
   \]
   So, the constant rate of change is \( \frac{\Delta k}{\Delta j} = -2.3 \).

2. **Fill in the missing values** (part b):

   Since we know the slope \( \frac{\Delta k}{\Delta j} = -2.3 \), we can use this to find the missing values in the table.

   - For \( j = 0 \), we need to find \( k \).
     Using the slope formula and the point \( (j = -2, k = 18.6) \):
     \[
     k - 18.6 = -2.3 \times (0 - (-2)) = -2.3 \times 2 = -4.6
     \]
     \[
     k = 18.6 - 4.6 = 14.0
     \]
     So, when \( j = 0 \), \( k = 14.0 \).

   - For \( j = -7 \), we also need to find \( k \).
     Using the slope formula and the point \( (j = -2, k = 18.6) \):
     \[
     k - 18.6 = -2.3 \times (-7 - (-2)) = -2.3 \times (-5) = 11.5
     \]
     \[
     k = 18.6 + 11.5 = 30.1
     \]
     So, when \( j = -7 \), \( k = 30.1 \).

   Updated table:
   | \( j \) | \( k \) |
   |--------|--------|
   | -7     | 30.1   |
   | -2     | 18.6   |
   | 0      | 14.0   |
   | ?      | 7.1    |
   | 6      | 0.2    |

   The missing \( j \)-value corresponding to \( k = 7.1 \) can be found using the equation for a line, which leads to \( j = 3 \).

3. **Formula for the linear relationship** (part c):

   The equation of a line is generally in the form:
   \[
   k = m j + b
   \]
   where \( m \) is the slope, and \( b \) is the y-intercept (the value of \( k \) when \( j = 0 \)).

   From our calculation, the slope \( m = -2.3 \) and the intercept \( b = 14.0 \).

   Thus, the equation is:
   \[
   k = -2.3 j + 14.0
   \]

Would you like more details on any step, or explanations on how to generalize this to other linear functions?

Here are 5 related questions for deeper understanding:
1. How can we verify the linearity of a dataset by calculating the slope between all points?
2. Can the slope between any two points in the table be different if the relationship is linear?
3. How can we predict values outside the provided \( j \)-range using the equation?
4. What is the significance of the slope and intercept in real-world scenarios?
5. How can we graph the relationship between \( j \) and \( k \) to visualize the linearity?

**Tip:** Always calculate the slope between two known points before filling in missing values. This ensures consistency throughout the table.

Determine the constant rate of change, fill in the table with the missing values, and write the formula that represents the value of k with respect to j.

Learn how to determine the constant rate of change and find missing values in a linear table. This problem involves calculating the slope and forming the equation k = mj + b, providing a clear understanding of linear relationships for students in Grades 8-10.

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