If [tex]\( k \)[/tex] is a nonzero constant and line [tex]\( I \)[/tex] in the [tex]\( xy \)[/tex]-plane passes through the points [tex]\((k, 2k)\)[/tex] and [tex]\((2k, 3k)\)[/tex], which of the following is an equation for line [tex]\( I \)[/tex]?

A. [tex]\( y = x + k \)[/tex]

B. [tex]\( y = 3x - k \)[/tex]

C. [tex]\( y = kx \)[/tex]

D. [tex]\( y = kx - k \)[/tex]



Answer :

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].