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 determine which equation represents a line that passes through [tex]\(\left(4, \frac{1}{3}\right)\)[/tex] with a slope of [tex]\(\frac{3}{4}\)[/tex], we can use the point-slope form of a linear equation. 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]\(\left(4, \frac{1}{3}\right)\)[/tex] implies [tex]\(x_1 = 4\)[/tex] and [tex]\(y_1 = \frac{1}{3}\)[/tex].
- Slope [tex]\(m = \frac{3}{4}\)[/tex].

Substitute these values into the point-slope form:

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

Now let's compare this equation with the given options:

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

The only option that matches our derived equation [tex]\(y - \frac{1}{3} = \frac{3}{4}(x - 4)\)[/tex] is the second option.

Therefore, the correct equation is:

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

Hence, the correct option is [tex]\( \boxed{2} \)[/tex].