To determine whether the given pair of lines are parallel, perpendicular, or neither, we need to consider the slopes of the lines.
The equations of the lines are:
1. [tex]\( y = 2x + 4 \)[/tex]
2. [tex]\( x - 2y = -5 \)[/tex]
Step 1: Identify the slope of the first line.
The first line is already in slope-intercept form [tex]\( y = mx + b \)[/tex] where [tex]\( m \)[/tex] is the slope.
For the first line, [tex]\( y = 2x + 4 \)[/tex], the slope (m1) is [tex]\( 2 \)[/tex].
Step 2: Convert the second line to slope-intercept form [tex]\( y = mx + b \)[/tex].
Starting with the second equation:
[tex]\[ x - 2y = -5 \][/tex]
First, solve for [tex]\( y \)[/tex]:
[tex]\[ -2y = -x - 5 \][/tex]
[tex]\[ y = \frac{-x - 5}{-2} \][/tex]
[tex]\[ y = \frac{1}{2}x + \frac{5}{2} \][/tex]
So, the second line in slope-intercept form is:
[tex]\[ y = 0.5x + 2.5 \][/tex]
The slope (m2) of this line is [tex]\( 0.5 \)[/tex].
Step 3: Compare the slopes to determine the relationship between the lines.
- If the slopes are equal ([tex]\( m_1 = m_2 \)[/tex]), the lines are parallel.
- If the slopes are negative reciprocals of each other ([tex]\( m_1 \cdot m_2 = -1 \)[/tex]), the lines are perpendicular.
- If neither condition is met, the lines are neither parallel nor perpendicular.
For our lines:
- [tex]\( m_1 = 2 \)[/tex]
- [tex]\( m_2 = 0.5 \)[/tex]
Now check the conditions:
- [tex]\( m_1 \neq m_2 \)[/tex], so the lines are not parallel.
- [tex]\( m_1 \cdot m_2 = 2 \cdot 0.5 = 1 \neq -1 \)[/tex], so the lines are not perpendicular.
Since neither condition for parallel or perpendicular is met, the lines are neither parallel nor perpendicular.
Answer:
C. neither
