To find the equation of a line parallel to [tex]\( y = 5x + 2 \)[/tex] that passes through the point [tex]\((-6, -1)\)[/tex], we need to follow these steps:
1. Identify the slope of the original line:
The given line is in the slope-intercept form [tex]\( y = mx + c \)[/tex], where [tex]\( m \)[/tex] is the slope. For the equation [tex]\( y = 5x + 2 \)[/tex], the slope [tex]\( m \)[/tex] is 5.
2. Determine the slope of the parallel line:
Two lines are parallel if they have the same slope. Therefore, the slope of the required parallel line is also 5.
3. Use the point-slope form to find the equation:
The point-slope form of a line's equation 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.
Given the point [tex]\((-6, -1)\)[/tex] and the slope [tex]\( 5 \)[/tex], we can substitute these values into the point-slope form:
[tex]\[
y - (-1) = 5(x - (-6))
\][/tex]
Simplifying, we get:
[tex]\[
y + 1 = 5(x + 6)
\][/tex]
4. Expand and simplify to get the slope-intercept form:
Next, simplify the equation:
[tex]\[
y + 1 = 5x + 30
\][/tex]
Subtract 1 from both sides to solve for [tex]\( y \)[/tex]:
[tex]\[
y = 5x + 29
\][/tex]
Thus, the equation of the line parallel to [tex]\( y = 5x + 2 \)[/tex] that passes through the point [tex]\((-6, -1)\)[/tex] is:
[tex]\[
y = 5x + 29
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{y = 5x + 29}
\][/tex]
From the given choices, the answer is:
[tex]\[ \boxed{B} \][/tex]
