To determine which equation represents the point-slope form of the line passing through the point [tex]\((3, 2)\)[/tex] with a slope of [tex]\(\frac{1}{3}\)[/tex], follow these steps:
1. Understand the point-slope form equation: The point-slope form of a line's 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.
2. Identify the known values: Here, we are given:
- A point on the line [tex]\((x_1, y_1) = (3, 2)\)[/tex]
- The slope [tex]\(m = \frac{1}{3}\)[/tex]
3. Substitute the known values into the point-slope form equation:
[tex]\[
y - 2 = \frac{1}{3}(x - 3)
\][/tex]
4. Compare 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]
Upon comparison, [tex]\(y - 2 = \frac{1}{3}(x - 3)\)[/tex] matches exactly with our derived equation.
Thus, 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]
So, the correct equation is: [tex]\(y - 2 = \frac{1}{3}(x - 3)\)[/tex].
