To determine which equation represents a line that passes through the point [tex]\((4, \frac{1}{3})\)[/tex] and has a slope of [tex]\(\frac{3}{4}\)[/tex], we can start by using the point-slope form of the equation of a line. The point-slope form is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\((x_1, y_1)\)[/tex] is the given point and [tex]\(m\)[/tex] is the slope.
Given:
[tex]\[ (x_1, y_1) = \left(4, \frac{1}{3}\right) \][/tex]
[tex]\[ m = \frac{3}{4} \][/tex]
Substituting these values into the point-slope form, we get:
[tex]\[ y - \frac{1}{3} = \frac{3}{4}(x - 4) \][/tex]
Now, let's compare this 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 equation [tex]\( y - \frac{1}{3} = \frac{3}{4}(x - 4) \)[/tex] exactly matches the equation derived from the point-slope form.
Therefore, the correct equation is:
[tex]\[ y - \frac{1}{3} = \frac{3}{4}(x - 4) \][/tex]
Hence, the correct answer is option 2.
