To solve this problem, let's go through the steps methodically to find the equation of the line that is parallel to the given line and passes through the point [tex]\( (-3, 1) \)[/tex].
### Step 1: Understand the problem requirements
We are given a line equation in point-slope form: [tex]\( y - 1 = (3/2)(x + 3) \)[/tex]. We need to find the equation of a line that is:
- Parallel to this given line.
- Passes through the point [tex]\( (-3, 1) \)[/tex].
### Step 2: Determine the slope of the given line
The general point-slope form of a line is [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( (x_1, y_1) \)[/tex] is a point on the line.
By comparing the given line's equation [tex]\( y - 1 = \frac{3}{2}(x + 3) \)[/tex] to the general form:
- We can identify that the slope [tex]\( m \)[/tex] is [tex]\( \frac{3}{2} \)[/tex].
### Step 3: Determine the slope of the parallel line
Parallel lines have the same slope. Therefore, the line parallel to the given line will also have a slope of [tex]\( \frac{3}{2} \)[/tex].
### Step 4: Use the point-slope form to find the equation of the parallel line
We will now use the point [tex]\( (-3, 1) \)[/tex] that the line passes through.
Substitute:
- [tex]\( x_1 = -3 \)[/tex]
- [tex]\( y_1 = 1 \)[/tex]
- [tex]\( m = \frac{3}{2} \)[/tex]
into the point-slope form equation [tex]\( y - y_1 = m(x - x_1) \)[/tex]:
[tex]\[ y - 1 = \frac{3}{2}(x - (-3)) \][/tex]
[tex]\[ y - 1 = \frac{3}{2}(x + 3) \][/tex]
### Step 5: Match the equation with the options provided
From the derived equation [tex]\( y - 1 = \frac{3}{2}(x + 3) \)[/tex], we can see that this matches the fourth option provided:
[tex]\[
\boxed{y-1=\frac{3}{2}(x+3)}
\][/tex]
So, the correct answer is the fourth option, [tex]\( y-1=\frac{3}{2}(x+3) \)[/tex].
