To determine whether the lines are perpendicular, parallel, or neither, we need to analyze their slopes. Let's first convert each equation to the slope-intercept form, [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] represents the slope.
1. For the first equation [tex]\( 6x - 2y = -2 \)[/tex]:
- Rewrite it to solve for [tex]\( y \)[/tex]:
[tex]\[
6x - 2y = -2
\][/tex]
[tex]\[
-2y = -6x - 2
\][/tex]
[tex]\[
y = 3x + 1
\][/tex]
- The slope [tex]\( m_1 \)[/tex] of the first line is [tex]\( 3 \)[/tex].
2. The second equation, [tex]\( y = 3x + 12 \)[/tex], is already in slope-intercept form.
- The slope [tex]\( m_2 \)[/tex] of the second line is also [tex]\( 3 \)[/tex].
Now, we compare the slopes [tex]\( m_1 \)[/tex] and [tex]\( m_2 \)[/tex]:
- If the slopes are equal, the lines are parallel.
- If the product of the slopes is [tex]\(-1\)[/tex], the lines are perpendicular.
- If neither condition is met, the lines are neither parallel nor perpendicular.
Here, the slopes are:
- [tex]\( m_1 = 3 \)[/tex]
- [tex]\( m_2 = 3 \)[/tex]
Since [tex]\( m_1 = m_2 \)[/tex]:
- The lines are parallel.
Therefore, we can complete the statement as follows:
The comparison of their slopes is equal, so the lines are parallel.
