To find the equation of a line in point-slope form that passes through the point [tex]\((3, 2)\)[/tex] and has a slope of [tex]\(\frac{1}{3}\)[/tex], follow these steps:
1. Identify the given point: The given point the line passes through is [tex]\((3, 2)\)[/tex]. Here, [tex]\(x_1 = 3\)[/tex] and [tex]\(y_1 = 2\)[/tex].
2. Identify the given slope: The slope [tex]\(m\)[/tex] is [tex]\(\frac{1}{3}\)[/tex].
3. Write the point-slope form equation: The point-slope form of the equation of a line is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Substituting the given point and slope into this equation:
[tex]\[
y - 2 = \frac{1}{3}(x - 3)
\][/tex]
4. Verify the point-slope equation: We need to check which of the given options matches this form.
- 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]
The correct equation that matches [tex]\(y - 2 = \frac{1}{3}(x - 3)\)[/tex] is option 2.
Therefore, the correct option is:
[tex]\[
\boxed{2}
\][/tex]
