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]
[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]
