To find the values of [tex]\(k\)[/tex] and [tex]\(n\)[/tex], and ultimately [tex]\(k + n\)[/tex], we will use the information provided about the points on the line and the given slope.
The slope of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is calculated using the formula:
[tex]\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Given the points [tex]\((3, 7)\)[/tex] and [tex]\((12, n)\)[/tex] with a slope of 2, we use the slope formula to find [tex]\(n\)[/tex]:
[tex]\[
2 = \frac{n - 7}{12 - 3}
\][/tex]
Simplify the denominator:
[tex]\[
2 = \frac{n - 7}{9}
\][/tex]
To solve for [tex]\(n\)[/tex], multiply both sides by 9:
[tex]\[
2 \times 9 = n - 7 \implies 18 = n - 7
\][/tex]
Add 7 to both sides:
[tex]\[
n = 18 + 7 \implies n = 25
\][/tex]
Now, let's find the value of [tex]\(k\)[/tex]. Given the points [tex]\((3, 7)\)[/tex] and [tex]\((k, 11)\)[/tex] with the same slope of 2, we use the slope formula again:
[tex]\[
2 = \frac{11 - 7}{k - 3}
\][/tex]
Simplify the numerator:
[tex]\[
2 = \frac{4}{k - 3}
\][/tex]
To solve for [tex]\(k\)[/tex], multiply both sides by [tex]\((k - 3)\)[/tex]:
[tex]\[
2(k - 3) = 4
\][/tex]
Distribute the 2:
[tex]\[
2k - 6 = 4
\][/tex]
Add 6 to both sides:
[tex]\[
2k = 10
\][/tex]
Divide by 2:
[tex]\[
k = 5
\][/tex]
Now that we have [tex]\(k = 5\)[/tex] and [tex]\(n = 25\)[/tex], we find [tex]\(k + n\)[/tex]:
[tex]\[
k + n = 5 + 25 = 30
\][/tex]
Therefore, the value of [tex]\(k + n\)[/tex] is [tex]\(\boxed{30}\)[/tex].
