Answer :
To determine which equation represents the 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 line equation. 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.
Here, \((x_1, y_1) = \left(4, \frac{1}{3}\right)\) and \(m = \frac{3}{4}\).
Substituting these values into the point-slope form equation, we get:
[tex]\[ y - \frac{1}{3} = \frac{3}{4}(x - 4) \][/tex]
Looking through the options given:
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. \( y - 4 = \frac{3}{4}\left(x - \frac{1}{3}\right) \)
Clearly, option 2 matches our derived equation exactly:
[tex]\[ y - \frac{1}{3} = \frac{3}{4}(x - 4) \][/tex]
Thus:
The equation that represents a line that passes through \(\left(4, \frac{1}{3}\right)\) and has a slope of \(\frac{3}{4}\) is:
[tex]\[ y - \frac{1}{3} = \frac{3}{4}(x - 4) \][/tex]
Therefore, the correct option is:
[tex]\[ y - \frac{1}{3} = \frac{3}{4}(x - 4) \][/tex]
which corresponds to option 2.
[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.
Here, \((x_1, y_1) = \left(4, \frac{1}{3}\right)\) and \(m = \frac{3}{4}\).
Substituting these values into the point-slope form equation, we get:
[tex]\[ y - \frac{1}{3} = \frac{3}{4}(x - 4) \][/tex]
Looking through the options given:
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. \( y - 4 = \frac{3}{4}\left(x - \frac{1}{3}\right) \)
Clearly, option 2 matches our derived equation exactly:
[tex]\[ y - \frac{1}{3} = \frac{3}{4}(x - 4) \][/tex]
Thus:
The equation that represents a line that passes through \(\left(4, \frac{1}{3}\right)\) and has a slope of \(\frac{3}{4}\) is:
[tex]\[ y - \frac{1}{3} = \frac{3}{4}(x - 4) \][/tex]
Therefore, the correct option is:
[tex]\[ y - \frac{1}{3} = \frac{3}{4}(x - 4) \][/tex]
which corresponds to option 2.
