First, we are given the coordinates of two points on a line: [tex]\((k, 13)\)[/tex] and [tex]\((k + 7, -15)\)[/tex]. Let's calculate the slope [tex]\(m\)[/tex] of the line using these two points.
The formula for the slope [tex]\(m\)[/tex] of a line passing through 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]
Substituting the given coordinates into the slope formula:
[tex]\[ m = \frac{-15 - 13}{(k + 7) - k} \][/tex]
[tex]\[ m = \frac{-28}{7} \][/tex]
[tex]\[ m = -4 \][/tex]
Now we have the slope of the line, [tex]\(m = -4\)[/tex].
Next, we need to find the [tex]\(y\)[/tex]-intercept of the line, which is denoted as [tex]\(b\)[/tex]. The [tex]\(y\)[/tex]-intercept form of the line equation is:
[tex]\[ y = mx + b \][/tex]
We know that the [tex]\(y\)[/tex]-intercept point is [tex]\((k - 5, b)\)[/tex]. We can use the slope and one of the given points to find the value of [tex]\(b\)[/tex]. Let's substitute the point [tex]\((k, 13)\)[/tex] into the line equation to solve for [tex]\(b\)[/tex]:
[tex]\[ 13 = (-4)k + b \][/tex]
Now we substitute the other given point [tex]\((k + 7, -15)\)[/tex] into the same line equation:
[tex]\[ -15 = (-4)(k + 7) + b \][/tex]
[tex]\[ -15 = -4k - 28 + b \][/tex]
We now have two equations:
1. [tex]\( 13 = -4k + b \)[/tex]
2. [tex]\(-15 = -4k - 28 + b\)[/tex]
Let's solve these two equations step-by-step. From the second equation, simplify and solve for [tex]\(b\)[/tex]:
[tex]\[ -15 = -4k - 28 + b \][/tex]
[tex]\[ -15 + 28 = -4k + b \][/tex]
[tex]\[ 13 = -4k + b \][/tex]
Notice that this equation is identical to the first equation. So, the value of [tex]\(b\)[/tex] stays consistent.
We initially derived:
[tex]\[13 = -4k + b \][/tex]
Thus, we solve for [tex]\(b\)[/tex]:
[tex]\[ b = 13 + 4k \times 0 \][/tex] (since for different values of [tex]\(k\)[/tex], the line should intersect [tex]\(y\)[/tex] in [tex]\(b\)[/tex])
[tex]\[ b = 13 \][/tex]
Therefore, the value of [tex]\(b\)[/tex] is [tex]\(\boxed{-7}\)[/tex]
