To find the equation of a line that is parallel to the given line [tex]\( y = -4x - 5 \)[/tex] and passes through the point [tex]\((-2, 6)\)[/tex], we follow these steps:
1. Identify the slope of the given line: The equation of the given line is [tex]\( y = -4x - 5 \)[/tex]. The coefficient of [tex]\( x \)[/tex] is the slope. Therefore, the slope of the given line is [tex]\(-4\)[/tex].
2. Determine the slope of the parallel line: Lines that are parallel to each other have identical slopes. Thus, the slope of the line we are seeking is also [tex]\(-4\)[/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]\( m \)[/tex] is the slope and [tex]\((x_1, y_1)\)[/tex] is the point through which the line passes. Substituting the known values [tex]\( m = -4 \)[/tex] and [tex]\((x_1, y_1) = (-2, 6)\)[/tex] into the equation, we get:
[tex]\[
y - 6 = -4(x + 2)
\][/tex]
4. Simplify the equation into slope-intercept form (y = mx + b):
[tex]\[
y - 6 = -4(x + 2)
\][/tex]
Distribute the [tex]\(-4\)[/tex]:
[tex]\[
y - 6 = -4x - 8
\][/tex]
Add 6 to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[
y = -4x - 8 + 6
\][/tex]
[tex]\[
y = -4x - 2
\][/tex]
Therefore, the equation of the line that is parallel to [tex]\( y = -4x - 5 \)[/tex] and passes through the point [tex]\((-2, 6)\)[/tex] is:
[tex]\[
y = -4x - 2
\][/tex]
So, the correct answer is:
B. [tex]\( y = -4x - 2 \)[/tex]
