To find the equation of a line, consider its slope and a point through which it passes. The line should be perpendicular to the given line [tex]\( y = -\frac{1}{2} x - 1 \)[/tex] and contain the point [tex]\( (2, 4) \)[/tex].
### Step-by-Step Solution:
1. Determine the slope of the given line:
The given line is [tex]\( y = -\frac{1}{2} x - 1 \)[/tex]. The slope [tex]\( m \)[/tex] of this line is [tex]\( -\frac{1}{2} \)[/tex].
2. Find the slope of the perpendicular line:
For two lines to be perpendicular, the product of their slopes must be [tex]\(-1\)[/tex]. Mathematically, if [tex]\( m_1 \)[/tex] is the slope of the first line, and [tex]\( m_2 \)[/tex] is the slope of the line perpendicular to it, then:
[tex]\[
m_1 \cdot m_2 = -1
\][/tex]
Given [tex]\( m_1 = -\frac{1}{2} \)[/tex]:
[tex]\[
-\frac{1}{2} \cdot m_2 = -1
\][/tex]
Solving for [tex]\( m_2 \)[/tex]:
[tex]\[
m_2 = 2
\][/tex]
Therefore, the slope of the perpendicular line is [tex]\( 2 \)[/tex].
3. Use the point-slope form of the equation of a line:
The perpendicular line must pass through the point [tex]\( (2, 4) \)[/tex]. The point-slope form of the equation is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Here, [tex]\( (x_1, y_1) = (2, 4) \)[/tex] and [tex]\( m = 2 \)[/tex]:
[tex]\[
y - 4 = 2(x - 2)
\][/tex]
4. Simplify to obtain the equation in slope-intercept form:
Distribute the slope on the right side:
[tex]\[
y - 4 = 2x - 4
\][/tex]
Add 4 to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[
y = 2x
\][/tex]
Thus, the equation of the line that is perpendicular to [tex]\( y = -\frac{1}{2} x - 1 \)[/tex] and passes through the point [tex]\( (2, 4) \)[/tex] is:
[tex]\[
\boxed{y = 2x}
\][/tex]
