To determine which equation represents a line that passes through the point [tex]\(\left(4, \frac{1}{3}\right)\)[/tex] and has a slope of [tex]\(\frac{3}{4}\)[/tex], we need to use 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 a point on the line and [tex]\(m\)[/tex] is the slope.
Given:
- Point: [tex]\((x_1, y_1) = \left(4, \frac{1}{3}\right)\)[/tex]
- Slope: [tex]\(m = \frac{3}{4}\)[/tex]
We substitute these values into the point-slope form equation:
[tex]\[ y - \frac{1}{3} = \frac{3}{4}(x - 4) \][/tex]
Thus, the correct equation that represents the line passing through the point [tex]\(\left(4, \frac{1}{3}\right)\)[/tex] with slope [tex]\(\frac{3}{4}\)[/tex] is:
[tex]\[ y - \frac{1}{3} = \frac{3}{4}(x - 4) \][/tex]
Therefore, the correct option among the given choices is:
[tex]\[ \boxed{2} \][/tex]
Or explicitly:
[tex]\[ y-\frac{1}{3}=\frac{3}{4}(x-4) \][/tex]
