Sure, let's go through the steps to find the equation of a line perpendicular to [tex]\( y = \frac{1}{2} x - 5 \)[/tex] that passes through the point [tex]\( (2, -7) \)[/tex].
1. Identify the slope of the given line:
The equation of the given line is in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. Here, the slope [tex]\( m \)[/tex] is [tex]\( \frac{1}{2} \)[/tex].
2. Determine the slope of the perpendicular line:
The slope of a line perpendicular to another is the negative reciprocal of the slope of the given line. Therefore, the slope of the perpendicular line is [tex]\( -\frac{1}{2} \)[/tex]. Taking the negative reciprocal involves flipping the fraction and changing the sign. Hence, the perpendicular slope is [tex]\( -2 \)[/tex].
3. Use the point-slope form equation:
The point-slope form of a line is given by the formula:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( (x_1, y_1) \)[/tex] is a point on the line. In this case, our point [tex]\( (x_1, y_1) \)[/tex] is [tex]\( (2, -7) \)[/tex] and the slope [tex]\( m \)[/tex] is [tex]\( -2 \)[/tex].
4. Plug in the slope and point into the point-slope form:
[tex]\[
y - (-7) = -2(x - 2)
\][/tex]
5. Simplify the equation:
Simplifying the left side, we get:
[tex]\[
y + 7 = -2(x - 2)
\][/tex]
This simplifies directly from the point-slope form to:
[tex]\[
y - (-7) = -2(x - 2)
\][/tex]
With all the steps completed, the equation of the line perpendicular to [tex]\( y = \frac{1}{2} x - 5 \)[/tex] passing through the point [tex]\( (2, -7) \)[/tex] in point-slope form is:
[tex]\[
y - (-7) = -2(x - 2)
\][/tex]
Therefore, the correct option is:
[tex]\[ y-(-7)=-2(x-2) \][/tex]
