To determine the correct symbol for the equation of the line that is perpendicular to [tex]\( y = -2x - 9 \)[/tex] and passes through the point [tex]\( (8, -4) \)[/tex], let's follow these steps:
1. Identify the slope of the given line:
The given line equation is [tex]\( y = -2x - 9 \)[/tex]. The slope of this line ([tex]\( m_1 \)[/tex]) is [tex]\(-2\)[/tex].
2. Find the slope of the perpendicular line:
The slope of a line that is perpendicular to another is the negative reciprocal of the original slope. If [tex]\( m_1 = -2 \)[/tex], the slope of the perpendicular line ([tex]\( m_2 \)[/tex]) is given by:
[tex]\[
m_2 = -\frac{1}{m_1} = -\frac{1}{-2} = \frac{1}{2}
\][/tex]
3. Use the point-slope form to find the equation of the perpendicular line:
We have a point [tex]\((x_1, y_1) = (8, -4)\)[/tex] and a slope [tex]\( m_2 = \frac{1}{2} \)[/tex].
The point-slope form of a line is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Substituting the given point and slope:
[tex]\[
y - (-4) = \frac{1}{2}(x - 8)
\][/tex]
Simplify this equation step by step:
[tex]\[
y + 4 = \frac{1}{2}(x - 8)
\][/tex]
Distribute the [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[
y + 4 = \frac{1}{2}x - 4
\][/tex]
Subtract 4 from both sides to solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{1}{2}x - 4 - 4
\][/tex]
[tex]\[
y = \frac{1}{2}x - 8
\][/tex]
4. Determine the correct symbol in the equation format [tex]\( y = [?] \frac{1}{2}x + [] \)[/tex]:
The final equation of the perpendicular line is [tex]\( y = \frac{1}{2}x - 8 \)[/tex].
The correct symbol in the green box is [tex]\( - \)[/tex].
