Let's analyze the given equations step by step to determine the relationship between the lines.
### Step 1: Rewrite the equations in slope-intercept form (y = mx + b)
#### Equation 1: [tex]\( 6x = 2y - 2 \)[/tex]
First, we simplify and rearrange the equation to isolate [tex]\( y \)[/tex]:
[tex]\[ 6x + 2 = 2y \][/tex]
Next, we divide each term by 2 to solve for [tex]\( y \)[/tex]:
[tex]\[ 3x + 1 = y \][/tex]
This is now in slope-intercept form [tex]\( (y = mx + b) \)[/tex], where the slope [tex]\( m \)[/tex] is 3.
#### Equation 2: [tex]\( y = 3x + 12 \)[/tex]
This equation is already in the slope-intercept form [tex]\( y = mx + b \)[/tex].
The slope [tex]\( m \)[/tex] is 3.
### Step 2: Compare the slopes
We have determined that:
- The slope (m) of the first line is 3.
- The slope (m) of the second line is 3.
### Step 3: Determine the relationship between the lines
To understand the relationship:
- If the slopes [tex]\( m_1 \)[/tex] and [tex]\( m_2 \)[/tex] are equal, then the lines are parallel.
- If the product of slopes [tex]\( m_1 \times m_2 \)[/tex] equals -1, then the lines are perpendicular.
- If neither of these conditions is met, the lines are neither parallel nor perpendicular.
Given that both slopes are 3 ([tex]\( m_1 = 3 \)[/tex] and [tex]\( m_2 = 3 \)[/tex]):
- The slopes are equal.
Therefore, the lines are parallel.
In conclusion:
The slope of their slopes is 3 and 3, so the lines are parallel.
