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The problem you're dealing with involves finding the value of \( b \), where \( b \) is the y-intercept of a line passing through two given points. Let's break down the information.

### Given Information:
The table provides the coordinates of two points on the line:
1. \( (k, 13) \)
2. \( (k + 7, -15) \)

We are also given that the y-intercept of the line is \( (k - 5, b) \), where \( k \) and \( b \) are constants, and the goal is to find the value of \( b \).

### Steps:
1. **Find the slope** of the line passing through the two given points.
   - The formula for the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
   \[
   m = \frac{y_2 - y_1}{x_2 - x_1}
   \]
   Substituting the coordinates \( (k, 13) \) and \( (k+7, -15) \):
   \[
   m = \frac{-15 - 13}{(k+7) - k} = \frac{-28}{7} = -4
   \]

2. **Use the point-slope form** of the line equation to express the equation of the line. The point-slope form is:
   \[
   y - y_1 = m(x - x_1)
   \]
   Using the point \( (k, 13) \) and slope \( m = -4 \):
   \[
   y - 13 = -4(x - k)
   \]
   Expanding this equation:
   \[
   y - 13 = -4x + 4k
   \]
   So, the equation of the line is:
   \[
   y = -4x + 4k + 13
   \]

3. **Find the y-intercept** \( b \).
   The y-intercept occurs when \( x = k - 5 \). Substituting \( x = k - 5 \) into the equation of the line:
   \[
   y = -4(k - 5) + 4k + 13
   \]
   Simplifying:
   \[
   y = -4k + 20 + 4k + 13 = 33
   \]
   So, the y-intercept is \( b = 33 \).

### Final Answer:
The value of \( b \) is **33**.

Would you like more details or have any questions?

### Here are 5 related questions:
1. How do you derive the point-slope form of a line?
2. Can you explain how the slope is used in determining the y-intercept?
3. What are other ways to find the equation of a line given two points?
4. How do you find the equation of a line if one point and the slope are known?
5. What happens to the line equation if \( k = 0 \)?

### Tip:
Remember that the slope represents how steep the line is, and the y-intercept tells you where the line crosses the y-axis!

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?

Learn how to find the y-intercept (b) of a line passing through two points in the xy-plane. This problem covers key concepts such as slope calculation and the point-slope form of a line. Suitable for students in Grades 10-12.

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