To determine which equation represents a line that passes through [tex]\(\left(4, \frac{1}{3}\right)\)[/tex] with a slope of [tex]\(\frac{3}{4}\)[/tex], we can use the point-slope form of a linear equation. The point-slope form 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.
Given:
- Point [tex]\(\left(4, \frac{1}{3}\right)\)[/tex] implies [tex]\(x_1 = 4\)[/tex] and [tex]\(y_1 = \frac{1}{3}\)[/tex].
- Slope [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]
Now let's compare this equation with the given options:
1. [tex]\(y - \frac{3}{4} = \frac{1}{3}(x - 4)\)[/tex]
2. [tex]\(y - \frac{1}{3} = \frac{3}{4}(x - 4)\)[/tex]
3. [tex]\(y - \frac{1}{3} = 4\left(x - \frac{3}{4}\right)\)[/tex]
4. [tex]\(y - 4 = \frac{3}{4}\left(x - \frac{1}{3}\right)\)[/tex]
The only option that matches our derived equation [tex]\(y - \frac{1}{3} = \frac{3}{4}(x - 4)\)[/tex] is the second option.
Therefore, the correct equation is:
[tex]\[ y - \frac{1}{3} = \frac{3}{4}(x - 4) \][/tex]
Hence, the correct option is [tex]\( \boxed{2} \)[/tex].
