To find the equation of a line that passes through a point and has a given slope, we use the point-slope form of the equation of a line, which is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here:
- [tex]\( (x_1, y_1) \)[/tex] is a point on the line.
- [tex]\( m \)[/tex] is the slope of the line.
In this problem, the given point is [tex]\( (3, 2) \)[/tex] and the slope [tex]\( m \)[/tex] is [tex]\( \frac{1}{3} \)[/tex].
Plugging these values into the point-slope form formula:
[tex]\[ y - 2 = \frac{1}{3}(x - 3) \][/tex]
So, the equation of the line in point-slope form that passes through the point [tex]\( (3, 2) \)[/tex] with a slope of [tex]\( \frac{1}{3} \)[/tex] is:
[tex]\[ y - 2 = \frac{1}{3}(x - 3) \][/tex]
Now, let's match this equation with the given options:
- [tex]\( y + 2 = \frac{1}{3}(x + 3) \)[/tex]
- [tex]\( y - 2 = \frac{1}{3}(x - 3) \)[/tex]
- [tex]\( y + 3 = \frac{1}{3}(x + 2) \)[/tex]
- [tex]\( y - 3 = \frac{1}{3}(x - 2) \)[/tex]
Clearly, the equation [tex]\( y - 2 = \frac{1}{3}(x - 3) \)[/tex] matches the form we derived.
Therefore, the correct equation is:
[tex]\[ y - 2 = \frac{1}{3}(x - 3) \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{y - 2 = \frac{1}{3}(x - 3)} \][/tex]
