To find the equation of a line that is parallel to a given line and passes through a specific point, we follow these steps:
1. Identify the slope of the given line:
- The equation of the given line is [tex]\( y - 1 = -\frac{3}{2}(x + 3) \)[/tex].
- This equation is in point-slope form [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\( m \)[/tex] is the slope.
- From the equation, we can see that the slope [tex]\( m \)[/tex] is [tex]\( -\frac{3}{2} \)[/tex].
2. Use the same slope for the parallel line:
- Parallel lines have the same slope. Therefore, the slope of our line will also be [tex]\( -\frac{3}{2} \)[/tex].
3. Use the point-slope form to write the equation:
- We are given a point [tex]\( (-3, 1) \)[/tex] through which our line passes.
- The point-slope form of the equation of a line is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
- Substituting [tex]\( m = -\frac{3}{2} \)[/tex], [tex]\( x_1 = -3 \)[/tex], and [tex]\( y_1 = 1 \)[/tex] into the formula, we get:
[tex]\[
y - 1 = -\frac{3}{2}(x - (-3))
\][/tex]
4. Simplify the equation:
- Simplifying the equation, we have:
[tex]\[
y - 1 = -\frac{3}{2}(x + 3)
\][/tex]
Therefore, the equation of the line in point-slope form that is parallel to the given line and passes through the point [tex]\((-3, 1)\)[/tex] is:
[tex]\[
y - 1 = -\frac{3}{2}(x + 3)
\][/tex]
So, the correct answer is:
[tex]\[
y - 1 = -\frac{3}{2}(x + 3)
\][/tex]
