Let's determine the relationship between the lines based on their slopes. Here are the given equations:
[tex]\[
6x - 2y = -2
\][/tex]
[tex]\[
y = 3x + 12
\][/tex]
First, we need to convert the first equation to the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
Starting with the equation:
[tex]\[
6x - 2y = -2
\][/tex]
We isolate [tex]\( y \)[/tex]:
[tex]\[
-2y = -6x - 2
\][/tex]
Divide by -2:
[tex]\[
y = 3x + 1
\][/tex]
From this, we can see that the slope ([tex]\( m \)[/tex]) of the line is 3.
For the second equation:
[tex]\[
y = 3x + 12
\][/tex]
This equation is already in slope-intercept form, and we can identify the slope ([tex]\( m \)[/tex]) of this line as 3.
Now, we compare the slopes of both lines:
- Slope of the first line: 3
- Slope of the second line: 3
The slopes of these lines are equal (both are 3), which means the lines are parallel.
Thus, we can conclude:
The slopes of their slopes is 3, so the lines are parallel.
