To solve the problem of finding an equation in point-slope form for the line that is parallel to [tex]\( y = \frac{1}{2} x - 7 \)[/tex] and passes through the point [tex]\((-3, -2)\)[/tex], we need to follow a few steps.
### Step 1: Identify the slope of the given line
The given line is [tex]\( y = \frac{1}{2} x - 7 \)[/tex]. The slope-intercept form of a line is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. Therefore, the slope of the given line is [tex]\( \frac{1}{2} \)[/tex].
### Step 2: Use the slope of the parallel line
Lines that are parallel have the same slope. Therefore, the slope of the line we need to find is also [tex]\( \frac{1}{2} \)[/tex].
### Step 3: Use the point-slope formula
The point-slope formula of a line 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 of the line.
Given the point [tex]\((-3, -2)\)[/tex] and slope [tex]\( \frac{1}{2} \)[/tex], substitute these values into the point-slope formula:
[tex]\[ y - (-2) = \frac{1}{2}(x - (-3)) \][/tex]
### Step 4: Simplify the equation
Simplify the equation to match the standard point-slope form:
[tex]\[ y + 2 = \frac{1}{2}(x + 3) \][/tex]
Thus, the equation of the line in point-slope form that passes through [tex]\((-3, -2)\)[/tex] and is parallel to [tex]\( y = \frac{1}{2} x - 7 \)[/tex] is:
[tex]\[ y + 2 = \frac{1}{2}(x + 3) \][/tex]
### Step 5: Identify the correct choice
Compare the simplified equation with the options provided:
A. [tex]\( y + 3 = -\frac{1}{2}(x + 2) \)[/tex]
B. [tex]\( y + 2 = 2(x + 3) \)[/tex]
C. [tex]\( y + 2 = \frac{1}{2}(x + 3) \)[/tex]
D. [tex]\( y - 2 = \frac{1}{2}(x - 3) \)[/tex]
The correct option that matches our equation [tex]\( y + 2 = \frac{1}{2}(x + 3) \)[/tex] is:
C. [tex]\( y + 2 = \frac{1}{2}(x + 3) \)[/tex]
So the answer is C.
