To solve the problem, we need to determine the slope of the line, find the equation of the line in slope-intercept form, and then use this to find the [tex]\(y\)[/tex]-intercept when [tex]\(x = k-5\)[/tex].
Given points:
- Point 1: [tex]\((x_1, y_1) = (k, 13)\)[/tex]
- Point 2: [tex]\((x_2, y_2) = (k + 7, -15)\)[/tex]
### Step 1: Calculate the slope (m) of the line
The formula for the slope [tex]\(m\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substitute the given values:
[tex]\[
m = \frac{-15 - 13}{(k + 7) - k} = \frac{-28}{7} = -4
\][/tex]
The slope of the line is [tex]\(-4\)[/tex].
### Step 2: Write the equation of the line in point-slope form
The point-slope form of a line is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Using point 1 [tex]\((x_1, y_1) = (k, 13)\)[/tex] and the slope [tex]\(m = -4\)[/tex]:
[tex]\[
y - 13 = -4(x - k)
\][/tex]
Simplify and rearrange to get the slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[
y - 13 = -4x + 4k
\][/tex]
[tex]\[
y = -4x + 4k + 13
\][/tex]
So, the equation of the line is:
[tex]\[
y = -4x + 4k + 13
\][/tex]
### Step 3: Find the [tex]\(y\)[/tex]-intercept for [tex]\(x = k - 5\)[/tex]
To find the value of [tex]\(b\)[/tex] when [tex]\(x = k-5\)[/tex]:
[tex]\[
y = -4(k - 5) + 4k + 13
\][/tex]
Simplify the expression:
[tex]\[
y = -4k + 20 + 4k + 13
\][/tex]
[tex]\[
y = 20 + 13
\][/tex]
[tex]\[
y = 33
\][/tex]
So, the [tex]\(y\)[/tex]-intercept of the line when [tex]\(x = k-5\)[/tex] is [tex]\(33\)[/tex].
Thus, the value of [tex]\(b\)[/tex] is [tex]\(\boxed{33}\)[/tex].
