To find the equation of a line that is parallel to a given line and passes through a specific point, we can 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].
- By comparing this equation with the point-slope form [tex]\( y - y_1 = m(x - x_1) \)[/tex], we see that the slope [tex]\( m \)[/tex] of the given line is [tex]\( -\frac{3}{2} \)[/tex].
2. Determine the slope of the parallel line:
- Lines that are parallel have the same slope. Therefore, the slope of the line parallel to the given line is also [tex]\( -\frac{3}{2} \)[/tex].
3. Use the point-slope form of the equation to find the equation of the parallel line passing through the point [tex]\((-3,1)\)[/tex]:
- The point-slope form is given by [tex]\( y - y_1 = m(x - x_1) \)[/tex].
- Here, [tex]\( m = -\frac{2}{3} \)[/tex], [tex]\( x_1 = -3 \)[/tex], and [tex]\( y_1 = 1 \)[/tex].
4. Substitute the values into the point-slope form:
- Using the given point [tex]\((-3,1)\)[/tex]:
[tex]\[
y - 1 = -\frac{2}{3}(x - (-3))
\][/tex]
- Simplifying the expression inside the parentheses:
[tex]\[
y - 1 = -\frac{2}{3}(x + 3)
\][/tex]
Therefore, the equation of the line that is parallel to the given line and passes through the point [tex]\((-3,1)\)[/tex] is:
[tex]\[
y - 1 = -\frac{2}{3}(x + 3)
\][/tex]
The correct answer is:
[tex]\[
y-1=-\frac{2}{3}(x+3)
\][/tex]
