To determine the equation of the line [tex]\(I\)[/tex] that passes through the points [tex]\((k, 2k)\)[/tex] and [tex]\((2k, 3k)\)[/tex], we need to find the slope and the equation of the line in slope-intercept form, [tex]\(y = mx + b\)[/tex].
Step 1: Find the slope [tex]\(m\)[/tex]
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 given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Using the points [tex]\((k, 2k)\)[/tex] and [tex]\((2k, 3k)\)[/tex]:
[tex]\[
m = \frac{3k - 2k}{2k - k} = \frac{k}{k} = 1
\][/tex]
Step 2: Use the point-slope form to find the equation
The point-slope form of the equation of a line is:
[tex]\[
y - y_1 = m (x - x_1)
\][/tex]
We can use point [tex]\((k, 2k)\)[/tex]:
[tex]\[
y - 2k = 1 \cdot (x - k)
\][/tex]
Simplifying:
[tex]\[
y - 2k = x - k
\][/tex]
Step 3: Solve for [tex]\(y\)[/tex] to get it in slope-intercept form
Add [tex]\(2k\)[/tex] to both sides to isolate [tex]\(y\)[/tex]:
[tex]\[
y = x - k + 2k
\][/tex]
[tex]\[
y = x + k
\][/tex]
Thus, the equation of line [tex]\(I\)[/tex] in slope-intercept form is:
[tex]\[
y = x + k
\][/tex]
Hence, the equation for line [tex]\(I\)[/tex] is:
[tex]\[
\boxed{y = x + k}
\][/tex]
This matches the given choice [tex]\(y = x + k\)[/tex].
