To find the point-slope form of the line passing through the point [tex]\((3, 2)\)[/tex] with a slope of [tex]\(\frac{1}{3}\)[/tex], we use the point-slope formula for a line, which is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here, [tex]\( (x_1, y_1) \)[/tex] represents the coordinates of the given point, and [tex]\( m \)[/tex] represents the slope of the line.
Given:
[tex]\[ x_1 = 3, \][/tex]
[tex]\[ y_1 = 2, \][/tex]
[tex]\[ m = \frac{1}{3} \][/tex]
We substitute these values into the point-slope formula:
[tex]\[ y - 2 = \frac{1}{3}(x - 3) \][/tex]
This is the equation of the line in point-slope form.
Now, let's match this equation with the given choices:
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]
From our analysis:
The correct equation from the given choices is:
[tex]\[ y - 2 = \frac{1}{3}(x - 3) \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{2} \][/tex]
