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We are given two points on a line, as shown in the table:

- Point 1: \((k, 13)\)
- Point 2: \((k+7, -15)\)

The equation also states that the **y-intercept** of the line is at \((k - 5, b)\), where \(k\) and \(b\) are constants, and we are tasked with finding \(b\).

### Steps to find \(b\):

1. **Find the slope of the line (m):**
   Using the formula for slope between two points \((x_1, y_1)\) and \((x_2, y_2)\):

   \[
   m = \frac{y_2 - y_1}{x_2 - x_1}
   \]

   Substituting the given points \((k, 13)\) and \((k+7, -15)\):

   \[
   m = \frac{-15 - 13}{(k + 7) - k} = \frac{-28}{7} = -4
   \]

   So, the slope \(m = -4\).

2. **Use the slope-intercept form of the line equation:**
   The equation of the line is \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.

   We can now use one of the points to find the y-intercept. Using the point \((k, 13)\), we substitute \(x = k\), \(y = 13\), and \(m = -4\) into the equation \(y = mx + c\):

   \[
   13 = -4k + c
   \]

   Solving for \(c\):

   \[
   c = 13 + 4k
   \]

   So, the equation of the line is:

   \[
   y = -4x + 13 + 4k
   \]

3. **Find the value of \(b\):**
   The y-intercept of the line occurs when \(x = k - 5\), so we substitute \(x = k - 5\) into the equation of the line:

   \[
   b = -4(k - 5) + 13 + 4k
   \]

   Simplifying:

   \[
   b = -4k + 20 + 13 + 4k
   \]

   \[
   b = 33
   \]

Thus, the value of \(b\) is **33**.

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Would you like more details on any part of the solution, or have any questions?

Here are 5 related questions:
1. How do you calculate the slope between two points?
2. What is the slope-intercept form of a linear equation?
3. How do you find the y-intercept given a point on a line and the slope?
4. What does the slope represent in a line's equation?
5. Can you explain how to use two points to write the full equation of a line?

**Tip:** Always ensure to substitute carefully when using given points to avoid small algebraic errors!

The table gives the coordinates of two points on a line in the xy-plane. The y-intercept of the line is (k - 5, b), where k and b are constants. What is the value of b?

Solve a problem where two points on a line are given, and the task is to find the y-intercept. This solution includes concepts of slope and linear equations, suitable for students in Grade 9-10.

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