Let's begin by using the point-slope form of a linear equation to determine which given option is correct.
### Step-by-Step Explanation:
1. Point-Slope Form Equation:
The point-slope form of a line that passes through a point [tex]\((x_1, y_1)\)[/tex] with a slope [tex]\(m\)[/tex] is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
2. Substitute the Given Values:
We are given a point [tex]\((4, \frac{1}{3})\)[/tex] and a slope [tex]\(\frac{3}{4}\)[/tex].
- Here, [tex]\(x_1 = 4\)[/tex]
- [tex]\(y_1 = \frac{1}{3}\)[/tex]
- [tex]\(m = \frac{3}{4}\)[/tex]
Substitute these values into the point-slope form:
[tex]\[
y - \frac{1}{3} = \frac{3}{4}(x - 4)
\][/tex]
3. Matching with Given Options:
Now, we compare this equation with the provided options to find the matching one.
- Option 1: [tex]\(y - \frac{3}{4} = \frac{1}{3}(x - 4)\)[/tex]
This does not match because the constants and the slope are different.
- Option 2: [tex]\(y - \frac{1}{3} = \frac{3}{4}(x - 4)\)[/tex]
This matches exactly with our derived equation!
- Option 3: [tex]\(y - \frac{1}{3} = 4\left(x - \frac{3}{4}\right)\)[/tex]
This does not match because the slope is not [tex]\(\frac{3}{4}\)[/tex], and the constants are incorrect.
- Option 4: [tex]\(y - 4 = \frac{3}{4}\left(x - \frac{1}{3}\right)\)[/tex]
This does not match because the constants are incorrect and the point doesn't match.
Thus, the correct option is:
[tex]\[
\boxed{2} \quad y - \frac{1}{3} = \frac{3}{4}(x - 4)
\][/tex]
