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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]\(\boxed{\text{relationship}}\)[/tex] of their slopes is [tex]\(\boxed{\text{perpendicular, parallel, or neither}}\)[/tex], so the lines are [tex]\(\boxed{\text{perpendicular, parallel, or neither}}\)[/tex].



Answer :

GinnyAnswer
To determine the relationship between the lines, we first need to find the slopes of both lines.

### Step 1: Find the slope of the first line
The first equation given is:
[tex]\[ 6x - 2y = -2 \][/tex]

To find the slope, we need to convert this equation into slope-intercept form, [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] represents the slope.

1. Add [tex]\( 2y \)[/tex] to both sides:
[tex]\[ 6x = 2y - 2 \][/tex]

2. Add 2 to both sides:
[tex]\[ 6x + 2 = 2y \][/tex]

3. Divide both sides by 2:
[tex]\[ 3x + 1 = y \][/tex]

Now that we have it in the form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y = 3x + 1 \][/tex]
The slope ([tex]\( m_1 \)[/tex]) of the first line is 3.

### Step 2: Find the slope of the second line
The second equation given is:
[tex]\[ y = 3x + 12 \][/tex]

The slope ([tex]\( m_2 \)[/tex]) of the second line is directly given as 3.

### Step 3: Determine the relationship between the lines
Now, we compare the slopes of the two lines.

- If two lines are parallel, their slopes are equal.
- If two lines are perpendicular, the product of their slopes is -1.

We have:
[tex]\[ m_1 = 3 \][/tex]
[tex]\[ m_2 = 3 \][/tex]

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

### Final Answer
The slope of their slopes is equal (both are 3), so the lines are parallel.

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