To determine the equation in point-slope form of the line passing through the point [tex]\((3, 2)\)[/tex] with a slope of [tex]\(\frac{1}{3}\)[/tex], we follow these steps:
1. Recall the point-slope form of a line equation:
The point-slope form is given by:
[tex]\[
y - y_1 = m (x - x_1)
\][/tex]
where [tex]\( (x_1, y_1) \)[/tex] is a point on the line and [tex]\( m \)[/tex] is the slope.
2. Substitute the given point and slope into the point-slope form equation:
Here, the point provided is [tex]\((x_1, y_1) = (3, 2)\)[/tex], and the slope ([tex]\(m\)[/tex]) is given as [tex]\(\frac{1}{3}\)[/tex]. Substituting these values in, we get:
[tex]\[
y - 2 = \frac{1}{3}(x - 3)
\][/tex]
3. Examine the given choices:
Now we look at the possible equations provided to determine which one corresponds to our derived equation [tex]\( y - 2 = \frac{1}{3}(x - 3) \)[/tex]. The choices given are:
1. [tex]\( y + 2 = \frac{1}{3}(x + 3) \)[/tex]
2. [tex]\( y - 2 = \frac{1}{3}(x - 3) \)[/tex]
3. [tex]\( y + 3 = \frac{1}{3}(x + 2) \)[/tex]
4. [tex]\( y - 3 = \frac{1}{3}(x - 2) \)[/tex]
Clearly, choice 2 matches our derived equation:
[tex]\[
y - 2 = \frac{1}{3}(x - 3)
\][/tex]
Therefore, the correct equation showing the point-slope form of the line that passes through [tex]\((3, 2)\)[/tex] and has a slope of [tex]\(\frac{1}{3}\)[/tex] is:
[tex]\[
y - 2 = \frac{1}{3}(x - 3)
\][/tex]
Thus, the correct choice is:
[tex]\[
\boxed{y - 2 = \frac{1}{3}(x - 3)}
\][/tex]
