To determine which line is perpendicular to line EF that has the equation [tex]\( y = -2x + 7 \)[/tex], we need to follow a series of steps. Here’s how to do it:
1. Identify the Slope of Line EF:
The equation of line EF is given as [tex]\( y = -2x + 7 \)[/tex]. This is in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- Here, the slope [tex]\( m \)[/tex] of line EF is [tex]\( -2 \)[/tex].
2. Determine the Slope of the Perpendicular Line:
Two lines are perpendicular if the product of their slopes is [tex]\( -1 \)[/tex]. This can be written as:
[tex]\[
m_1 \times m_2 = -1
\][/tex]
where [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.
- Given that the slope [tex]\( m_1 \)[/tex] of line EF is [tex]\( -2 \)[/tex], we need to find [tex]\( m_2 \)[/tex] such that:
[tex]\[
-2 \times m_2 = -1
\][/tex]
Solving for [tex]\( m_2 \)[/tex], we get:
[tex]\[
m_2 = \frac{-1}{-2} = \frac{1}{2}
\][/tex]
Hence, the slope of the line perpendicular to EF is [tex]\( \frac{1}{2} \)[/tex].
3. Evaluate the Given Options:
We now match this slope with the slopes of the given options:
- For the line [tex]\( y = 2x - 3 \)[/tex], the slope is [tex]\( 2 \)[/tex].
- For the line [tex]\( y = \frac{1}{2}x - 3 \)[/tex], the slope is [tex]\( \frac{1}{2} \)[/tex].
- For the line [tex]\( y = -2x - 3 \)[/tex], the slope is [tex]\( -2 \)[/tex].
- For the line [tex]\( y = -\frac{1}{2}x - 3 \)[/tex], the slope is [tex]\( -\frac{1}{2} \)[/tex].
Comparing these slopes with [tex]\( \frac{1}{2} \)[/tex], we see that the line [tex]\( y = \frac{1}{2}x - 3 \)[/tex] has the required slope.
Therefore, the equation of the line that is perpendicular to line EF is:
[tex]\[
y = \frac{1}{2}x - 3
\][/tex]
