To determine the relationship between the two given lines, we'll first find the slopes of both lines.
Step 1: Extract the slope from the first line.
The first equation is:
[tex]\[ 6x - 2y = -2 \][/tex]
To find its slope, we rearrange it into the slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[ 6x - 2y = -2 \][/tex]
[tex]\[ -2y = -6x - 2 \][/tex]
[tex]\[ y = 3x + 1\][/tex]
From the equation [tex]\(y = 3x + 1\)[/tex], we see that the slope ([tex]\(m\)[/tex]) is:
[tex]\[ \text{slope1} = 3 \][/tex]
Step 2: Extract the slope from the second line.
The second equation is already in slope-intercept form:
[tex]\[ y = 3x + 12 \][/tex]
From this equation, the slope ([tex]\(m\)[/tex]) is:
[tex]\[ \text{slope2} = 3 \][/tex]
Step 3: Compare the slopes of the two lines.
The slopes of both lines are:
[tex]\[ \text{slope1} = 3 \][/tex]
[tex]\[ \text{slope2} = 3 \][/tex]
Since the slopes are equal ([tex]\( \text{slope1} = \text{slope2} \)[/tex]), the lines are parallel.
Thus, the correct answers for the drop-down menus are:
"The ___ of their slopes is ___, so the lines are parallel ___."
- The relationship of their slopes is equal, so the lines are parallel always.
