To determine the equation of the line that is perpendicular to the given line [tex]\( y = 2x + 3 \)[/tex] and passes through the point [tex]\((-4, 8)\)[/tex], we will follow these steps:
1. Identify the slope of the given line:
The given equation is in slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. For the line [tex]\( y = 2x + 3 \)[/tex], the slope [tex]\( m \)[/tex] is 2.
2. Determine the slope of the perpendicular line:
The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope. Therefore, the slope [tex]\( m \)[/tex] of the perpendicular line is [tex]\(- \frac{1}{2}\)[/tex].
3. Use the point-slope form of the equation of a line:
The point-slope form of the equation of a line is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Here, [tex]\((x_1, y_1)\)[/tex] is a point on the line and [tex]\( m \)[/tex] is the slope. We know that the perpendicular line passes through the point [tex]\((-4, 8)\)[/tex] and its slope is [tex]\(-\frac{1}{2}\)[/tex]. Substituting these values into the point-slope form gives:
[tex]\[
y - 8 = -\frac{1}{2}(x + 4)
\][/tex]
4. Simplify the equation to slope-intercept form:
We now simplify the equation to get it in the form [tex]\( y = mx + b \)[/tex]:
[tex]\[
y - 8 = -\frac{1}{2}(x + 4)
\][/tex]
[tex]\[
y - 8 = -\frac{1}{2}x - 2
\][/tex]
[tex]\[
y = -\frac{1}{2}x - 2 + 8
\][/tex]
[tex]\[
y = -\frac{1}{2}x + 6
\][/tex]
Hence, 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]\[
\boxed{y = -\frac{1}{2} x + 6}
\][/tex]
