To determine if the lines are perpendicular, parallel, or neither, we first need to find the slopes of both lines. We will begin by rewriting each equation in slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] represents the slope.
1. Find the slope of the first line [tex]\(6x - 2y = -2\)[/tex]:
- Start by isolating [tex]\(y\)[/tex]:
[tex]\[
6x - 2y = -2
\][/tex]
- Subtract [tex]\(6x\)[/tex] from both sides:
[tex]\[
-2y = -6x - 2
\][/tex]
- Divide both sides by [tex]\(-2\)[/tex]:
[tex]\[
y = 3x + 1
\][/tex]
Thus, the slope of the first line is [tex]\(3\)[/tex].
2. Find the slope of the second line [tex]\(y = 3x + 12\)[/tex]:
This line is already in slope-intercept form.
Thus, the slope of the second line is [tex]\(3\)[/tex].
3. Determine the relationship between the slopes:
- If the slopes are equal ([tex]\(m_1 = m_2\)[/tex]), the lines are parallel.
- If the slopes are negative reciprocals ([tex]\(m_1 \cdot m_2 = -1\)[/tex]), the lines are perpendicular.
- If neither condition is met, the lines are neither parallel nor perpendicular.
In this case, both slopes are equal to 3. Since the slopes are equal, the lines are parallel.
Therefore, the sentences can be completed as:
"The product of their slopes is 1, so the lines are parallel."
