To determine which of the options represents the equation of a line that is perpendicular to the given line [tex]\( y = -\frac{1}{8} x - 2 \)[/tex] and passes through the point [tex]\((-2, -3)\)[/tex], we need to follow these steps:
1. Determine the slope of the given line:
The equation of the given line is in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. So here, [tex]\( m = -\frac{1}{8} \)[/tex].
2. Find the slope of the perpendicular line:
Perpendicular lines have slopes that are negative reciprocals of each other. If the slope of the given line is [tex]\( m_1 = -\frac{1}{8} \)[/tex], then the slope of the perpendicular line [tex]\( m_2 \)[/tex] is:
[tex]\[
m_2 = -\frac{1}{m_1} = -\frac{1}{-\frac{1}{8}} = 8
\][/tex]
3. Use the point-slope form to find the equation of the line:
The point-slope form of a line's equation is [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\( (x_1, y_1) \)[/tex] is a point on the line.
We are given the point [tex]\((-2, -3)\)[/tex], and we have found the slope [tex]\( m_2 = 8 \)[/tex]. Plugging in these values:
[tex]\[
y - (-3) = 8(x - (-2))
\][/tex]
Simplify the equation:
[tex]\[
y + 3 = 8(x + 2)
\][/tex]
Expand and isolate [tex]\( y \)[/tex]:
[tex]\[
y + 3 = 8x + 16
\][/tex]
[tex]\[
y = 8x + 13
\][/tex]
4. Identify the correct option:
The equation we derived is [tex]\( y = 8x + 13 \)[/tex], which corresponds to option [tex]\( d \)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{d. \ y = 8x + 13} \][/tex]
