To solve for the equation of the line that is perpendicular to [tex]\( y = -\frac{1}{2} x - 5 \)[/tex] and passes through the point [tex]\( (2, 7) \)[/tex], we need to follow these steps:
1. Identify the slope of the given line:
The equation of the given line is [tex]\( y = -\frac{1}{2} x - 5 \)[/tex]. From this equation, we can see that the slope [tex]\( m_1 \)[/tex] of the given line is [tex]\( -\frac{1}{2} \)[/tex].
2. Determine the slope of the perpendicular line:
The slope of a line perpendicular to another line is the negative reciprocal of the slope of the given line. So, if the slope of the given line is [tex]\( m_1 = -\frac{1}{2} \)[/tex], then the slope [tex]\( m_2 \)[/tex] of the perpendicular line is:
[tex]\[
m_2 = -\frac{1}{m_1} = -\frac{1}{-\frac{1}{2}} = 2
\][/tex]
3. Use the point-slope form of the equation:
We have the slope [tex]\( m_2 = 2 \)[/tex] and the point [tex]\( (2, 7) \)[/tex]. The point-slope form of the equation of a line is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\( (x_1, y_1) \)[/tex] is the point [tex]\((2, 7)\)[/tex].
4. Substitute the slope and point into the point-slope form:
[tex]\[
y - 7 = 2(x - 2)
\][/tex]
5. Simplify the equation to slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[
y - 7 = 2(x - 2)
\][/tex]
[tex]\[
y - 7 = 2x - 4
\][/tex]
[tex]\[
y = 2x - 4 + 7
\][/tex]
[tex]\[
y = 2x + 3
\][/tex]
Therefore, the equation of the line perpendicular to [tex]\( y = -\frac{1}{2} x - 5 \)[/tex] and passing through the point [tex]\( (2, 7) \)[/tex] in slope-intercept form is:
[tex]\[
y = 2x + 3
\][/tex]
