To determine the equation of the line that is perpendicular to [tex]\( y = 4x + 6 \)[/tex] and passes through the point [tex]\( (8, -4) \)[/tex], follow these steps:
1. Find the slope of the given line:
The given line equation is [tex]\( y = 4x + 6 \)[/tex]. This is in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. Therefore, the slope of the given line is [tex]\( m = 4 \)[/tex].
2. Determine the slope of the perpendicular line:
The slope of a line perpendicular to another is the negative reciprocal of the given slope. Thus, the slope of the line perpendicular to the given line is [tex]\( m_{\text{perpendicular}} = -\frac{1}{m} = -\frac{1}{4} \)[/tex].
3. Use the point-slope form to write the equation of the perpendicular line:
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 the point the line passes through, and [tex]\( m \)[/tex] is the slope of the line. In this case, the point is [tex]\( (8, -4) \)[/tex] and the slope is [tex]\( -\frac{1}{4} \)[/tex].
4. Substitute the given point and the slope into the point-slope form:
[tex]\[
y - (-4) = -\frac{1}{4}(x - 8)
\][/tex]
Simplifying this equation step-by-step:
[tex]\[
y + 4 = -\frac{1}{4}(x - 8)
\][/tex]
Distribute the [tex]\( -\frac{1}{4} \)[/tex]:
[tex]\[
y + 4 = -\frac{1}{4}x + 2
\][/tex]
Subtract 4 from both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = -\frac{1}{4}x + 2 - 4
\][/tex]
Simplify the expression:
[tex]\[
y = -\frac{1}{4}x - 2
\][/tex]
Thus, the equation of the line that is perpendicular to [tex]\( y = 4x + 6 \)[/tex] and passes through the point [tex]\( (8, -4) \)[/tex] is [tex]\( y = -\frac{1}{4}x - 2 \)[/tex].
The correct choice is:
[tex]\[
\boxed{y = -\frac{1}{4}x - 2}
\][/tex]
