What is 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]?

A. [tex]\( y - 1 = -\frac{3}{2}(x + 3) \)[/tex]
B. [tex]\( y - 1 = -\frac{2}{3}(x + 3) \)[/tex]
C. [tex]\( y - 1 = \frac{2}{3}(x + 3) \)[/tex]
D. [tex]\( y - 1 = \frac{3}{2}(x + 3) \)[/tex]



Answer :

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]