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 can use the point-slope form of a linear equation.
The point-slope form of a line 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
- [tex]\(m\)[/tex] is the slope of the line
Given:
[tex]\[
(x_1, y_1) = \left(4, \frac{1}{3}\right)
\][/tex]
[tex]\[
m = \frac{3}{4}
\][/tex]
Substituting the given point and slope into the point-slope form, we get:
[tex]\[
y - \frac{1}{3} = \frac{3}{4}(x - 4)
\][/tex]
This equation matches the second option provided:
[tex]\[ y - \frac{1}{3} = \frac{3}{4}(x - 4) \][/tex]
Therefore, the correct equation representing the line that passes through [tex]\(\left(4, \frac{1}{3}\right)\)[/tex] with a slope of [tex]\(\frac{3}{4}\)[/tex] is:
[tex]\[ \boxed{y-\frac{1}{3}=\frac{3}{4}(x-4)} \][/tex]
