To find the equation of the line parallel to the given line [tex]\( y = -3x + \frac{1}{8} \)[/tex] and passing through the point [tex]\( (-3, 1) \)[/tex], follow these steps:
1. Identify the slope of the given line:
The given line is in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] represents the slope. Here, [tex]\( m = -3 \)[/tex]. So, the slope of the given line is [tex]\( -3 \)[/tex].
2. Determine the slope of the new line:
Since the new line is parallel to the given line, it will have the same slope. Thus, the slope of the new line is also [tex]\( -3 \)[/tex].
3. Use the point-slope form of the equation of a line:
The point-slope form is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\( (x_1, y_1) \)[/tex] is a point on the line and [tex]\( m \)[/tex] is the slope.
4. Substitute the slope and the given point [tex]\( (-3, 1) \)[/tex] into the point-slope form:
[tex]\[
y - 1 = -3(x - (-3))
\][/tex]
[tex]\[
y - 1 = -3(x + 3)
\][/tex]
5. Simplify the equation:
Distribute the slope [tex]\( -3 \)[/tex] on the right side:
[tex]\[
y - 1 = -3x - 9
\][/tex]
Add 1 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = -3x - 9 + 1
\][/tex]
[tex]\[
y = -3x - 8
\][/tex]
So, the equation of the line parallel to [tex]\( y = -3x + \frac{1}{8} \)[/tex] that passes through [tex]\( (-3, 1) \)[/tex] is:
[tex]\[
y = -3x - 8
\][/tex]
Checking the given options:
- [tex]\( y - 1 = \frac{1}{2}(x - 4) \)[/tex]
- [tex]\( y = -3x - 6 \)[/tex]
- [tex]\( y = \frac{1}{2}x + 7 \)[/tex]
- [tex]\( 3x + y = -8 \)[/tex]
None of these are algebraically equivalent to [tex]\( y = -3x - 8 \)[/tex], but we can rearrange the last option:
For [tex]\( 3x + y = -8 \)[/tex]:
[tex]\[
y = -8 - 3x
\][/tex]
[tex]\[
y = -3x - 8
\][/tex]
This matches our equation.
Therefore, the correct option is:
[tex]\[
3x + y = -8
\][/tex]
