To determine which equation correctly represents a line passing through the point [tex]\(\left(4, \frac{1}{3}\right)\)[/tex] with a slope of [tex]\(\frac{3}{4}\)[/tex], we use the point-slope form of the equation of a line. This form is given as:
[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]
Inserting these values into the point-slope form equation:
[tex]\[ y - \frac{1}{3} = \frac{3}{4} (x - 4) \][/tex]
Now, we need to compare this equation to the given options:
1. [tex]\( y - \frac{3}{4} = \frac{1}{3} (x - 4) \)[/tex]
This option does not match our form. The term [tex]\(\frac{3}{4}\)[/tex] does not align with the slope of [tex]\(\frac{1}{3}\)[/tex], nor does [tex]\(\frac{3}{4}\)[/tex] align with the point [tex]\(\frac{1}{3}\)[/tex]. Thus, this is incorrect.
2. [tex]\( y - \frac{1}{3} = \frac{3}{4} (x - 4) \)[/tex]
This option matches exactly with our derived form: [tex]\( y - \frac{1}{3} = \frac{3}{4} (x - 4) \)[/tex]. Therefore, this is correct.
3. [tex]\( y - \frac{1}{3} = 4 \left( x - \frac{3}{4} \right) \)[/tex]
This option does not use the correct slope [tex]\(\frac{3}{4}\)[/tex]. Instead, it has slope [tex]\(4\)[/tex] and a subtractive point [tex]\(\frac{3}{4}\)[/tex] in place of [tex]\(4\)[/tex]. Therefore, this is incorrect.
4. [tex]\( y - 4 = \frac{3}{4} \left( x - \frac{1}{3} \right) \)[/tex]
This option changes both the [tex]\(y\)[/tex]-term from [tex]\(\frac{1}{3}\)[/tex] to a subtractive point of [tex]\(4\)[/tex] and the [tex]\(x\)[/tex]-term to [tex]\(\frac{1}{3}\)[/tex]. It does not match our form. Thus, it is incorrect.
The correct equation that represents the line passing through [tex]\(\left(4, \frac{1}{3}\right)\)[/tex] with a slope of [tex]\(\frac{3}{4}\)[/tex] is:
[tex]\[ y - \frac{1}{3} = \frac{3}{4} (x - 4) \][/tex]
Thus, the correct option is:
[tex]\[ \boxed{2} \][/tex]
