Let's begin by identifying the slopes of each line.
### Step 1: Convert the First Equation to Slope-Intercept Form
The first equation is \( 6x - 2y = -2 \). To find the slope, we need to convert it to the slope-intercept form, \( y = mx + b \), where \( m \) is the slope.
1. Start with the equation:
[tex]\[ 6x - 2y = -2 \][/tex]
2. Isolate the \( y \)-term by subtracting \( 6x \) from both sides:
[tex]\[ -2y = -6x - 2 \][/tex]
3. Divide everything by \(-2\) to solve for \( y \):
[tex]\[ y = 3x + 1 \][/tex]
So, the slope \( m_1 \) of the first line is \( 3 \).
### Step 2: Identify the Slope of the Second Line
The second equation is \( y = 3x + 12 \). This is already in slope-intercept form \( y = mx + b \), where the slope \( m_2 \) is \( 3 \).
### Step 3: Determine the Relationship Between the Slopes
We compare the slopes \( m_1 \) and \( m_2 \):
- \( m_1 = 3 \)
- \( m_2 = 3 \)
1. If the slopes are equal (\( m_1 = m_2 \)), then the lines are parallel.
2. If the product of the slopes is \(-1\) (i.e., \( m_1 \times m_2 = -1 \)), the lines are perpendicular.
3. Otherwise, the lines are neither parallel nor perpendicular.
Since the slopes are equal (\( 3 = 3 \)), the lines are parallel.
### Conclusion
The product of their slopes is \( 9 \), which we don't actually need to figure out since the comparison indicated that both slopes are equal. Therefore, the lines are parallel.
So, the correct selections are:
- The product of their slopes is 9 ,
- so the lines are parallel.
