Sure, let's solve this step-by-step:
1. Identify the slope of the given line:
The equation of the given line is [tex]\( y = -0.5x + B \)[/tex]. Here, the slope ([tex]\(m\)[/tex]) of this line is -0.5.
2. Determine the slope of the perpendicular line:
The slope of a line perpendicular to another is the negative reciprocal of the original line's slope.
- The negative reciprocal of -0.5 is [tex]\( \frac{1}{0.5} = 2 \)[/tex].
3. Equation of the perpendicular line in point-slope form:
We are given that this perpendicular line passes through the point [tex]\((-1, 6)\)[/tex].
- The point-slope form of the equation of a line is [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\((x_1, y_1)\)[/tex] is the given point.
- Here, [tex]\( m = 2 \)[/tex], [tex]\( x_1 = -1 \)[/tex], and [tex]\( y_1 = 6 \)[/tex].
4. Substitute the known values into the point-slope form:
[tex]\[
y - 6 = 2(x - (-1))
\][/tex]
Simplify:
[tex]\[
y - 6 = 2(x + 1)
\][/tex]
5. Convert to slope-intercept form ([tex]\( y = mx + b \)[/tex]):
[tex]\[
y - 6 = 2x + 2
\][/tex]
Add 6 to both sides:
[tex]\[
y = 2x + 8
\][/tex]
So, the equation of the perpendicular line in slope-intercept form that passes through the point [tex]\((-1, 6)\)[/tex] is [tex]\( y = 2x + 8 \)[/tex].
Given the options:
- [tex]\( y = -0.5x - 5.5 \)[/tex]
- [tex]\( y = -0.5x + 5.5 \)[/tex]
- [tex]\( y = 2x + 13 \)[/tex]
- [tex]\( y = 2x - 13 \)[/tex]
None of these match our result of [tex]\( y = 2x + 8 \)[/tex]. It seems there might be a small discrepancy in the provided options, but the correct result according to our steps is indeed [tex]\( y = 2x + 8 \)[/tex].
