To determine which equation represents a line passing through the point [tex]\((2, -\frac{1}{2})\)[/tex] with a slope of 3, we start by considering the point-slope form of a linear equation:
[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:
[tex]\[ x_1 = 2 \][/tex]
[tex]\[ y_1 = -\frac{1}{2} \][/tex]
[tex]\[ m = 3 \][/tex]
Substituting these values into the point-slope form, we get:
[tex]\[ y - \left(-\frac{1}{2}\right) = 3(x - 2) \][/tex]
[tex]\[ y + \frac{1}{2} = 3(x - 2) \][/tex]
Now, let's examine the given options:
1. [tex]\( y - 2 = 3\left(x + \frac{1}{2}\right) \)[/tex]
2. [tex]\( y - 3 = 2\left(x + \frac{1}{2}\right) \)[/tex]
3. [tex]\( y + \frac{1}{2} = 3(x - 2) \)[/tex]
4. [tex]\( y + \frac{1}{2} = 2(x - 3) \)[/tex]
By comparing the simplified form [tex]\( y + \frac{1}{2} = 3(x - 2) \)[/tex] with the given options, we see that it matches option 3.
Therefore, the correct equation representing a line that passes through [tex]\((2, -\frac{1}{2})\)[/tex] and has a slope of 3 is:
[tex]\[ y + \frac{1}{2} = 3(x - 2) \][/tex]
