To find the equation, in point-slope form, of the line that is parallel to the given line and passes through the point [tex]\((-3,1)\)[/tex], let's follow these steps:
1. Identify the slope of the given line:
The given line is [tex]\(y - 1 = -\frac{3}{2}(x + 3)\)[/tex].
From this equation, we can see that the slope ([tex]\(m\)[/tex]) of the given line is [tex]\(-\frac{3}{2}\)[/tex].
2. Parallel lines have the same slope:
Therefore, the slope of the required line will also be [tex]\(-\frac{3}{2}\)[/tex].
3. Use the point-slope form of the equation of a line:
The point-slope form of a line is [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.
4. Substitute the given point [tex]\((-3,1)\)[/tex] and the slope [tex]\(-\frac{3}{2}\)[/tex] into the point-slope form:
[tex]\[
y - 1 = -\frac{3}{2}(x - (-3))
\][/tex]
Simplifying, we get:
[tex]\[
y - 1 = -\frac{3}{2}(x + 3)
\][/tex]
This is the equation of the line in point-slope form.
Therefore, the correct answer is:
[tex]\[
y - 1 = -\frac{3}{2}(x + 3)
\][/tex]
