To find the equation of a line that is parallel to [tex]\( y = -5x + 6 \)[/tex] and passes through the point [tex]\( (-4, -1) \)[/tex], follow these steps:
1. Identify the slope of the given line.
- The given line is [tex]\( y = -5x + 6 \)[/tex].
- For a line in the slope-intercept form [tex]\( y = mx + b \)[/tex], the slope is [tex]\( m \)[/tex].
- Therefore, the slope of the given line is [tex]\( -5 \)[/tex].
2. Recognize that parallel lines have the same slope.
- Any line parallel to [tex]\( y = -5x + 6 \)[/tex] will also have a slope of [tex]\( -5 \)[/tex].
3. Use the point-slope form to write the equation of the new line.
- The point-slope form of a line is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
- Here, [tex]\( m \)[/tex] is the slope and [tex]\( (x_1, y_1) \)[/tex] is the point through which the line passes.
- For our case, [tex]\( m = -5 \)[/tex] and the point is [tex]\( (-4, -1) \)[/tex].
4. Substitute the slope and the point into the point-slope form:
[tex]\[
y - (-1) = -5(x - (-4))
\][/tex]
- Simplify the equation:
[tex]\[
y + 1 = -5(x + 4)
\][/tex]
5. Expand and simplify to get the equation into slope-intercept form:
- Distribute the slope on the right side:
[tex]\[
y + 1 = -5x - 20
\][/tex]
- Subtract 1 from both sides to solve for [tex]\( y \)[/tex]:
[tex]\[
y = -5x - 21
\][/tex]
Therefore, the equation of the line that is parallel to [tex]\( y = -5x + 6 \)[/tex] and passes through the point [tex]\( (-4, -1) \)[/tex] is [tex]\( y = -5x - 21 \)[/tex].
So, the correct answer is:
D. [tex]\( y = -5x - 21 \)[/tex]
