To determine the equation of the line in point-slope form that passes through the point [tex]\((3, 2)\)[/tex] and has a slope of [tex]\(\frac{1}{3}\)[/tex], we follow these steps:
1. Understand the Point-Slope Form:
The point-slope form of a linear equation 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 of the line.
2. Substitute the Given Point and Slope:
Given the point [tex]\((x_1, y_1) = (3, 2)\)[/tex] and the slope [tex]\(m = \frac{1}{3}\)[/tex], substitute these values into the point-slope form:
[tex]\[
y - 2 = \frac{1}{3}(x - 3)
\][/tex]
3. Identify the Correct Option:
We need to compare this equation with the given options to identify which one correctly represents the point-slope form derived above.
- Option 1: [tex]\(y + 2 = \frac{1}{3}(x + 3)\)[/tex]
- Option 2: [tex]\(y - 2 = \frac{1}{3}(x - 3)\)[/tex]
- Option 3: [tex]\(y + 3 = \frac{1}{3}(x + 2)\)[/tex]
- Option 4: [tex]\(y - 3 = \frac{1}{3}(x - 2)\)[/tex]
4. Check Each Option:
- Option 1: [tex]\(y + 2 = \frac{1}{3}(x + 3)\)[/tex]
Substituting the point [tex]\((3, 2)\)[/tex] does not satisfy this equation.
- Option 2: [tex]\(y - 2 = \frac{1}{3}(x - 3)\)[/tex]
This matches exactly with the point-slope form derived. Therefore, it is the correct equation.
- Option 3: [tex]\(y + 3 = \frac{1}{3}(x + 2)\)[/tex]
Substituting the point [tex]\((3, 2)\)[/tex] does not satisfy this equation.
- Option 4: [tex]\(y - 3 = \frac{1}{3}(x - 2)\)[/tex]
Substituting the point [tex]\((3, 2)\)[/tex] does not satisfy this equation.
Therefore, the equation that shows the point-slope form of the line passing through [tex]\((3, 2)\)[/tex] with a slope of [tex]\(\frac{1}{3}\)[/tex] is:
[tex]\[
y - 2 = \frac{1}{3}(x - 3)
\][/tex]
Thus, the correct answer is:
[tex]\(\boxed{2}\)[/tex]
