Which equation represents a line that passes through [tex]\left(4, \frac{1}{3}\right)[/tex] and has a slope of [tex]\frac{3}{4}[/tex]?

A. [tex]y - \frac{3}{4} = \frac{1}{3}(x - 4)[/tex]
B. [tex]y - \frac{1}{3} = \frac{3}{4}(x - 4)[/tex]
C. [tex]y - \frac{1}{3} = 4\left(x - \frac{3}{4}\right)[/tex]
D. [tex]y - 4 = \frac{3}{4}\left(x - \frac{1}{3}\right)[/tex]



Answer :

To find the equation of a line that passes through the point \(\left(4, \frac{1}{3}\right)\) and has a slope of \(\frac{3}{4}\), we can 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 \((x_1, y_1)\) is a point on the line and \(m\) is the slope.

Given:
- The point \((x_1, y_1) = \left(4, \frac{1}{3}\right)\)
- The slope \(m = \frac{3}{4}\)

We can substitute these values into the point-slope form equation:

[tex]\[ y - \frac{1}{3} = \frac{3}{4} (x - 4) \][/tex]

This is the equation of the line in point-slope form.

Now, let’s compare this equation with the given options:

1. \( y - \frac{3}{4} = \frac{1}{3} (x - 4) \)
2. \( y - \frac{1}{3} = \frac{3}{4} (x - 4) \)
3. \( y - \frac{1}{3} = 4 \left( x - \frac{3}{4} \right) \)
4. \( y - 4 = \frac{3}{4} \left( x - \frac{1}{3} \right) \)

We observe that the correct equation is:

[tex]\[ y - \frac{1}{3} = \frac{3}{4} (x - 4) \][/tex]

Thus, the correct option is:

[tex]\[ \boxed{2} \][/tex]