To find the equation of a line that is parallel to a given line and passes through a specified point, follow these steps:
1. Identify the slope of the given line:
The given equation is [tex]\( y - 1 = -\frac{3}{2}(x + 3) \)[/tex]. This is in point-slope form, which is generally written as:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Here, [tex]\( m \)[/tex] represents the slope. From the given 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 for the new line is also [tex]\( -\frac{3}{2} \)[/tex].
3. Use the point-slope form of the equation:
We need to write the equation of a line that passes through the point [tex]\((-3, 1)\)[/tex] with the slope [tex]\( -\frac{3}{2} \)[/tex]. The point-slope form is:
[tex]\[
y - y_1 = m (x - x_1)
\][/tex]
Plug in the given point and the slope:
[tex]\[
y - 1 = -\frac{3}{2}(x + 3)
\][/tex]
Here, [tex]\( x_1 = -3 \)[/tex] and [tex]\( y_1 = 1 \)[/tex].
4. Verify the point-slope form with the given options:
[tex]\[
y - 1 = -\frac{3}{2}(x + 3)
\][/tex]
After following these steps, we conclude that the equation of the line 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]
Therefore, the correct choice is the first one:
[tex]\[ y - 1 = -\frac{3}{2}(x + 3) \][/tex]
