To identify the equation of the line that is perpendicular to [tex]\( y = \frac{1}{2} x - 7 \)[/tex] and passes through the point [tex]\( (4, -2) \)[/tex], follow these steps:
1. Determine the slope of the original line:
The given equation is in slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. Hence, the slope of the original line [tex]\( y = \frac{1}{2} x - 7 \)[/tex] is [tex]\( \frac{1}{2} \)[/tex].
2. Find 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]\( m_{\perpendicular} \)[/tex] of the perpendicular line will be:
[tex]\[
m_{\perpendicular} = -\frac{1}{\left(\frac{1}{2}\right)} = -2
\][/tex]
3. Use the point-slope form to find the equation:
The point-slope form of the equation of a line is given by:
[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. Given the point [tex]\( (4, -2) \)[/tex] and the slope [tex]\( -2 \)[/tex], plug these values into the point-slope form:
[tex]\[
y + 2 = -2(x - 4)
\][/tex]
4. Simplify the equation:
Simplify the equation step by step:
[tex]\[
y + 2 = -2x + 8
\][/tex]
[tex]\[
y = -2x + 6
\][/tex]
Thus, the equation of the line that is perpendicular to [tex]\( y = \frac{1}{2} x - 7 \)[/tex] and passes through the point [tex]\( (4, -2) \)[/tex] is:
[tex]\[
\boxed{y = -2x + 6}
\][/tex]
