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Are these lines perpendicular, parallel, or neither based on their slopes?

[tex]\[
\begin{array}{l}
6x - 2y = -2 \\
y = 3x + 12
\end{array}
\][/tex]

The [tex]\(\square\)[/tex] of their slopes is [tex]\(\square\)[/tex], so the lines are [tex]\(\square\)[/tex].



Answer :

GinnyAnswer
To determine whether the lines are perpendicular, parallel, or neither, we need to find and compare their slopes.

1. Find the slope of the first line:
[tex]\[ 6x - 2y = -2 \][/tex]
To find the slope, we need to rewrite this equation in slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.

First, isolate [tex]\(y\)[/tex]:
[tex]\[ 6x - 2y = -2 \\ -2y = -6x - 2 \][/tex]
Now divide everything by [tex]\(-2\)[/tex]:
[tex]\[ y = 3x + 1 \][/tex]
Therefore, the slope [tex]\(m_1\)[/tex] of the first line is [tex]\(3\)[/tex].

2. Find the slope of the second line:
[tex]\[ y = 3x + 12 \][/tex]
This is already in slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.

Therefore, the slope [tex]\(m_2\)[/tex] of the second line is [tex]\(3\)[/tex].

3. Compare the slopes:
- If the slopes are equal ([tex]\(m_1 = m_2\)[/tex]), the lines are parallel.
- If the product of the slopes is [tex]\(-1\)[/tex] ([tex]\(m_1 \cdot m_2 = -1\)[/tex]), the lines are perpendicular.
- Otherwise, the lines are neither parallel nor perpendicular.

The slopes found are:
[tex]\[ m_1 = 3 \quad \text{and} \quad m_2 = 3 \][/tex]

Since [tex]\(m_1 = m_2\)[/tex], the lines are parallel.

To fill in the blanks:
1. The comparison of their slopes is equal.
2. So the lines are parallel.

So the complete sentence is:
The comparison of their slopes is equal, so the lines are parallel.

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