Question

which equation represents a line that passes through ((2, -\frac{1}{2})) and has a slope of 3?
(y - 3 = 2left(x + \frac{1}{2}
ight))
(y + \frac{1}{2} = 3(x - 2))
(y - 2 = 3left(x + \frac{1}{2}
ight))
(y + \frac{1}{2} = 2(x - 3))

Explanation:

Step1: Recall point - slope form

The point - slope form of a linear equation is given by \(y - y_1=m(x - x_1)\), where \((x_1,y_1)\) is a point on the line and \(m\) is the slope of the line.

Step2: Identify \(x_1\), \(y_1\) and \(m\)

We are given that the line passes through the point \((2,-\frac{1}{2})\), so \(x_1 = 2\) and \(y_1=-\frac{1}{2}\). The slope \(m = 3\).

Step3: Substitute into point - slope form

Substitute \(x_1 = 2\), \(y_1=-\frac{1}{2}\) and \(m = 3\) into the point - slope formula \(y - y_1=m(x - x_1)\).
We get \(y-(-\frac{1}{2})=3(x - 2)\), which simplifies to \(y+\frac{1}{2}=3(x - 2)\).

Answer:

\(y+\frac{1}{2}=3(x - 2)\) (the second option)