To determine which equation represents a line that passes through the point \(\left(2, -\frac{1}{2}\right)\) and has a slope of 3, 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 of the line.
Given:
- The point \((x_1, y_1) = \left(2, -\frac{1}{2}\right)\)
- The slope \(m = 3\)
Substitute these values into the point-slope form:
[tex]\[ y - \left(-\frac{1}{2}\right) = 3(x - 2) \][/tex]
Simplify the left side of the equation:
[tex]\[ y + \frac{1}{2} = 3(x - 2) \][/tex]
This is the simplified form of the equation of the line that passes through \((2, -\frac{1}{2})\) with a slope of 3.
Thus, the correct option is:
[tex]\[ y + \frac{1}{2} = 3(x - 2) \][/tex]
Therefore, the answer is:
[tex]\[ y + \frac{1}{2} = 3(x - 2) \][/tex]
