To determine 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], we need to follow several steps. Let's break it down:
1. Identify the slope of the given line:
The given line is in point-slope form:
[tex]\[ y - 1 = -\frac{3}{2}(x + 3) \][/tex]
The slope-intercept form of a line is generally [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. From this point-slope equation, we can see directly that the slope [tex]\( m = -\frac{3}{2} \)[/tex].
2. Understand that parallel lines have the same slope:
Since we are looking for a line that is parallel to the given one, it must have the same slope. Therefore, the slope of our desired line is also [tex]\( -\frac{3}{2} \)[/tex].
3. Write the equation using the point-slope form:
The point-slope form is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here, [tex]\((x_1, y_1)\)[/tex] is a point through which the line passes, and [tex]\( m \)[/tex] is the slope.
4. Insert the given point [tex]\((-3, 1)\)[/tex] and the slope [tex]\( -\frac{3}{2} \)[/tex]:
So, substituting [tex]\((x_1, y_1) = (-3, 1)\)[/tex] and [tex]\( m = -\frac{3}{2} \)[/tex] into the point-slope form equation, we get:
[tex]\[ y - 1 = -\frac{3}{2}(x - (-3)) \][/tex]
[tex]\[ y - 1 = -\frac{3}{2}(x + 3) \][/tex]
5. Find the correct answer:
The equation derived from the steps above is:
[tex]\[ y - 1 = -\frac{3}{2}(x + 3) \][/tex]
Therefore, the correct option is:
[tex]\[ y - 1 = -\frac{3}{2}(x + 3) \][/tex]
So, the answer to the question is:
[tex]\[ \boxed{1} \][/tex]
