Certainly! Let's analyze the given lines and determine their relationship based on their slopes.
We start with the given equations of the lines:
[tex]\[
\begin{array}{l}
6x - 2y = -2 \\
y = 3x + 12
\end{array}
\][/tex]
Step 1: Convert the first equation into slope-intercept form (y = mx + b)
First, we solve the equation [tex]\(6x - 2y = -2\)[/tex] for [tex]\(y\)[/tex]:
1. Isolate [tex]\(y\)[/tex] on one side:
[tex]\[
-2y = -6x - 2
\][/tex]
2. Divide by [tex]\(-2\)[/tex]:
[tex]\[
y = 3x + 1
\][/tex]
So, the slope-intercept form of the first equation is:
[tex]\[
y = 3x + 1
\][/tex]
Step 2: Identify the slopes of both lines
From the slope-intercept forms:
1. The first line is [tex]\(y = 3x + 1\)[/tex], so the slope [tex]\((m_1)\)[/tex] is [tex]\(3\)[/tex].
2. The second line is [tex]\(y = 3x + 12\)[/tex], so the slope [tex]\((m_2)\)[/tex] is [tex]\(3\)[/tex].
Step 3: Determine the relationship based on slopes
We compare the slopes:
1. If the slopes are equal [tex]\((m_1 = m_2)\)[/tex], the lines are parallel.
2. If the product of the slopes is [tex]\(-1\)[/tex] [tex]\((m_1 \cdot m_2 = -1)\)[/tex], the lines are perpendicular.
3. Otherwise, the lines are neither parallel nor perpendicular.
Since [tex]\(m_1 = 3\)[/tex] and [tex]\(m_2 = 3\)[/tex]:
[tex]\[
m_1 = m_2 = 3
\][/tex]
Therefore, the lines are parallel.
Now, we will select the correct answer from each drop-down menu based on this information:
1. The relationship of their slopes is:
[tex]\[
\text{equal}
\][/tex]
2. Since the slopes are equal, the lines are:
[tex]\[
\text{parallel}
\][/tex]
So, putting it all together:
The relationship of their slopes is equal, so the lines are parallel.
