To determine whether the lines are perpendicular, parallel, or neither, we need to find and compare their slopes.
1. Find the slope of the first line:
[tex]\[
6x - 2y = -2
\][/tex]
To find the slope, we need to rewrite this equation in slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
First, isolate [tex]\(y\)[/tex]:
[tex]\[
6x - 2y = -2 \\
-2y = -6x - 2
\][/tex]
Now divide everything by [tex]\(-2\)[/tex]:
[tex]\[
y = 3x + 1
\][/tex]
Therefore, the slope [tex]\(m_1\)[/tex] of the first line is [tex]\(3\)[/tex].
2. Find the slope of the second line:
[tex]\[
y = 3x + 12
\][/tex]
This is already in slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
Therefore, the slope [tex]\(m_2\)[/tex] of the second line is [tex]\(3\)[/tex].
3. Compare the slopes:
- If the slopes are equal ([tex]\(m_1 = m_2\)[/tex]), the lines are parallel.
- If the product of the slopes is [tex]\(-1\)[/tex] ([tex]\(m_1 \cdot m_2 = -1\)[/tex]), the lines are perpendicular.
- Otherwise, the lines are neither parallel nor perpendicular.
The slopes found are:
[tex]\[
m_1 = 3 \quad \text{and} \quad m_2 = 3
\][/tex]
Since [tex]\(m_1 = m_2\)[/tex], the lines are parallel.
To fill in the blanks:
1. The comparison of their slopes is equal.
2. So the lines are parallel.
So the complete sentence is:
The comparison of their slopes is equal, so the lines are parallel.
