To find the equation of a line that is perpendicular to [tex]\( y = 2x + 1 \)[/tex] and passes through the point [tex]\( (4,6) \)[/tex]:
1. Determine 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. Here, [tex]\( y = 2x + 1 \)[/tex] has a slope [tex]\( m = 2 \)[/tex].
2. Find the slope of the perpendicular line: The slope of a line perpendicular to another is the negative reciprocal of the original slope. The negative reciprocal of [tex]\( 2 \)[/tex] is [tex]\( -\frac{1}{2} \)[/tex].
3. Use the point-slope form of the equation of a line: The point-slope form is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\( (x_1, y_1) \)[/tex] is a point on the line and [tex]\( m \)[/tex] is the slope. Substitute [tex]\( m = -\frac{1}{2} \)[/tex] and the point [tex]\( (4, 6) \)[/tex].
4. Substitute the values into the point-slope form:
[tex]\[
y - 6 = -\frac{1}{2}(x - 4)
\][/tex]
5. Simplify the equation:
[tex]\[
y - 6 = -\frac{1}{2}x + 2
\][/tex]
Adding 6 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = -\frac{1}{2}x + 2 + 6
\][/tex]
[tex]\[
y = -\frac{1}{2}x + 8
\][/tex]
6. Determine the correct option: The equation we derived is [tex]\( y = -\frac{1}{2}x + 8 \)[/tex]. Among the given options, this matches the fourth option:
[tex]\[
\boxed{y = -\frac{1}{2}x + 8}
\][/tex]
Hence, the equation of the line that is perpendicular to [tex]\( y = 2x + 1 \)[/tex] and passes through the point [tex]\( (4, 6) \)[/tex] is [tex]\( y = -\frac{1}{2}x + 8 \)[/tex]. The correct answer is:
[tex]\[
\boxed{4}
\][/tex]
