To find the equation of the line that is perpendicular to [tex]\( y = 2x + 3 \)[/tex] and passes through the point [tex]\( (-4, 8) \)[/tex], follow these steps:
1. Identify the slope of the given line:
The given line is [tex]\( y = 2x + 3 \)[/tex]. The slope (m) of this line is 2.
2. Determine the slope of the perpendicular line:
The slope of a line that is perpendicular to another is the negative reciprocal of the original slope. The negative reciprocal of 2 is [tex]\( -\frac{1}{2} \)[/tex].
3. Use the point-slope form to find the y-intercept:
The point-slope form of a line equation is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Here, [tex]\( (x_1, y_1) \)[/tex] is the point [tex]\( (-4, 8) \)[/tex] and the slope (m) is [tex]\( -\frac{1}{2} \)[/tex].
Substitute the slope and the point into the equation:
[tex]\[
y - 8 = -\frac{1}{2}(x + 4)
\][/tex]
4. Simplify the equation:
Expand the right side:
[tex]\[
y - 8 = -\frac{1}{2} x - 2
\][/tex]
Add 8 to both sides to solve for y:
[tex]\[
y = -\frac{1}{2} x - 2 + 8
\][/tex]
Simplify the right side:
[tex]\[
y = -\frac{1}{2} x + 6
\][/tex]
Therefore, the equation of the line that is perpendicular to [tex]\( y = 2x + 3 \)[/tex] and passes through the point [tex]\( (-4, 8) \)[/tex] is:
[tex]\[
y = -\frac{1}{2} x + 6
\][/tex]
The correct answer is:
[tex]\[
y = -\frac{1}{2} x + 6
\][/tex]
