To solve for the equation of a line that is perpendicular to [tex]\( y = 4x + 5 \)[/tex] and passes through the point [tex]\((8, 3)\)[/tex], we need to follow these steps:
1. Find the slope of the perpendicular line:
The slope of a line perpendicular to another is the negative reciprocal of the slope of the original line.
The slope [tex]\( m_1 \)[/tex] of the original line [tex]\( y = 4x + 5 \)[/tex] is 4. Therefore, the slope [tex]\( m_2 \)[/tex] of the line perpendicular to it is:
[tex]\[
m_2 = -\frac{1}{4}
\][/tex]
2. Use the point-slope form to find the equation:
We use the point-slope form of a line’s equation:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\((x_1, y_1)\)[/tex] is a point on the line (which is (8, 3) in this case) and [tex]\( m \)[/tex] is the slope of the line.
Substituting in the known values:
[tex]\[
y - 3 = -\frac{1}{4} (x - 8)
\][/tex]
3. Simplify to get the slope-intercept form [tex]\( y = mx + b \)[/tex]:
Distribute the slope on the right side:
[tex]\[
y - 3 = -\frac{1}{4} x + 2
\][/tex]
Add 3 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = -\frac{1}{4} x + 2 + 3
\][/tex]
[tex]\[
y = -\frac{1}{4} x + 5
\][/tex]
Therefore, the equation of the line that is perpendicular to [tex]\( y = 4x + 5 \)[/tex] and passes through the point (8, 3) is:
[tex]\[
\boxed{y = -\frac{1}{4} x + 5}
\][/tex]
This matches option A. Therefore, the correct answer is:
[tex]\[
\boxed{A. \, y = -\frac{1}{4} x + 5}
\][/tex]
