To find the equation of a line in point-slope form 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 line is [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.
- For the given equation, the slope [tex]\( m \)[/tex] is [tex]\( -\frac{3}{2} \)[/tex].
2. Recognize that parallel lines have the same slope:
- Since the new line must be parallel to the given line, it will have the same slope [tex]\( m = -\frac{3}{2} \)[/tex].
3. Use the point-slope form of the equation of a line:
- The point-slope form is [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\( (x_1, y_1) \)[/tex] is a point on the line.
- The point given is [tex]\( (-3, 1) \)[/tex].
4. Substitute the point [tex]\((-3, 1)\)[/tex] and the slope [tex]\( -\frac{3}{2} \)[/tex] into the point-slope form:
- Begin by writing the formula: [tex]\( y - y_1 = m(x - x_1) \)[/tex].
- Replace [tex]\( y_1 \)[/tex] with 1, [tex]\( m \)[/tex] with [tex]\( -\frac{3}{2} \)[/tex], and [tex]\( x_1 \)[/tex] with [tex]\( -3 \)[/tex].
[tex]\[
y - 1 = -\frac{3}{2}(x - (-3))
\][/tex]
5. Simplify the equation:
- Simplify the expression inside the parenthesis:
[tex]\[
y - 1 = -\frac{3}{2}(x + 3)
\][/tex]
Therefore, 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] is:
[tex]\[
y - 1 = -\frac{3}{2}(x + 3)
\][/tex]
So the correct answer is:
[tex]\[
y - 1 = -\frac{3}{2}(x + 3)
\][/tex]
