To find the equation of 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 the equation of a line. The point-slope form of the equation 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, and [tex]\(m\)[/tex] is the slope.
1. Identify the given point [tex]\((x_1, y_1)\)[/tex]:
[tex]\[ x_1 = 4 \][/tex]
[tex]\[ y_1 = \frac{1}{3} \][/tex]
2. Identify the given slope [tex]\(m\)[/tex]:
[tex]\[ m = \frac{3}{4} \][/tex]
3. Substitute the given point and slope into the point-slope form equation:
[tex]\[ y - \frac{1}{3} = \frac{3}{4}(x - 4) \][/tex]
Thus, the equation representing the line that passes through [tex]\(\left(4, \frac{1}{3}\right)\)[/tex] and has a slope of [tex]\(\frac{3}{4}\)[/tex] is:
[tex]\[ y - \frac{1}{3} = \frac{3}{4}(x - 4) \][/tex]
Therefore, the correct option is:
[tex]\[ \boxed{y - \frac{1}{3} = \frac{3}{4}(x - 4)} \][/tex]
