Let's determine the equation of the line that passes through the point [tex]\((2, -\frac{1}{2})\)[/tex] and has a slope of 3.
The point-slope form of a linear equation 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.
Here, [tex]\((x_1, y_1) = (2, -\frac{1}{2})\)[/tex] and [tex]\(m = 3\)[/tex].
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 by adding [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[ y + \frac{1}{2} = 3(x - 2) \][/tex]
This is the simplified form of the equation. Now, let’s compare it to the provided options:
- [tex]\( y - 2 = 3 \left(x + \frac{1}{2}\right) \)[/tex]
- [tex]\( y - 3 = 2 \left(x + \frac{1}{2}\right) \)[/tex]
- [tex]\( y + \frac{1}{2} = 3(x - 2) \)[/tex]
- [tex]\( y + \frac{1}{2} = 2(x - 3) \)[/tex]
The correct form that matches our derived equation is:
[tex]\[ y + \frac{1}{2} = 3(x - 2) \][/tex]
Therefore, the correct equation from the options is:
[tex]\[ y + \frac{1}{2} = 3(x - 2) \][/tex]
So, the correct answer is:
[tex]\[ \boxed{3} \][/tex]
