To find the equation of the line that is perpendicular to the given line [tex]\( y = \frac{1}{4} x - 7 \)[/tex] and passes through the point [tex]\((-2, -6)\)[/tex], follow these steps:
1. Identify the slope of the given line:
The given line is in slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. For the line [tex]\( y = \frac{1}{4} x - 7 \)[/tex], the slope [tex]\( m \)[/tex] is [tex]\( \frac{1}{4} \)[/tex].
2. Determine the slope of the perpendicular line:
Two lines are perpendicular if the product of their slopes is [tex]\( -1 \)[/tex]. Therefore, the slope of the line perpendicular to [tex]\( \frac{1}{4} \)[/tex] is the negative reciprocal of [tex]\( \frac{1}{4} \)[/tex]:
[tex]\[
m_{\text{perpendicular}} = -\frac{1}{\frac{1}{4}} = -4
\][/tex]
3. Use the point-slope form of 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]\( m \)[/tex] is the slope and [tex]\((x_1, y_1)\)[/tex] is a point on the line.
4. Substitute the point [tex]\((-2, -6)\)[/tex] and the slope [tex]\(-4\)[/tex] into the point-slope equation:
[tex]\[
y - (-6) = -4(x - (-2))
\][/tex]
Thus, the equation in point-slope form for the line perpendicular to [tex]\( y = \frac{1}{4} x - 7 \)[/tex] and passing through the point [tex]\((-2, -6)\)[/tex] is:
[tex]\[
y - (-6) = -4(x - (-2))
\][/tex]
This simplifies to:
[tex]\[
y + 6 = -4(x + 2)
\][/tex]
So, the desired equation is:
[tex]\[
y - -6 = -4(x - -2)
\][/tex]
