Certainly! To determine the equation showing the point-slope form of the line that passes through the point [tex]\((3,2)\)[/tex] and has a slope of [tex]\(\frac{1}{3}\)[/tex], we can follow a systematic approach using the point-slope form of a line.
### Step-by-Step Solution
1. Recall the Point-Slope Form:
The point-slope form of a line is given by the formula:
[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 Values:
- The point we have is [tex]\((3, 2)\)[/tex], so [tex]\(x_1 = 3\)[/tex] and [tex]\(y_1 = 2\)[/tex].
- The slope [tex]\(m\)[/tex] is given as [tex]\(\frac{1}{3}\)[/tex].
Substitute these values into the point-slope form:
[tex]\[
y - 2 = \frac{1}{3}(x - 3)
\][/tex]
3. Match the Format:
Compare the obtained equation [tex]\(y - 2 = \frac{1}{3}(x - 3)\)[/tex] with the given choices:
- [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]
It is clear that the equation [tex]\(y - 2 = \frac{1}{3}(x - 3)\)[/tex] matches our derived equation.
### Conclusion
Hence, the correct equation showing the point-slope form of the line passing through [tex]\((3, 2)\)[/tex] with a slope of [tex]\(\frac{1}{3}\)[/tex] is
[tex]\[
\boxed{y - 2 = \frac{1}{3}(x - 3)}
\][/tex]
