Question

Are the Lines Parallel, Perpendicular, or Intersecting?

y=1/3x+2


2y+6x=4


(1 point)
Responses

Parallel
Parallel

Intersecting
Intersecting

Perpendicular
Perpendicular

Answer 1

To determine whether the lines are parallel, perpendicular, or intersecting, we first need to put both equations into slope-intercept form (y = mx + b), where m represents the slope.

1. The first equation is already in slope-intercept form:
\[
y = \frac{1}{3}x + 2
\]
Here, the slope (m₁) is \(\frac{1}{3}\).

2. The second equation is:
\[
2y + 6x = 4
\]
We can rearrange this into slope-intercept form:
\[
2y = -6x + 4
\]
\[
y = -3x + 2
\]
Here, the slope (m₂) is \(-3\).

Now we can compare the slopes:

- For the lines to be **parallel**, their slopes must be the same. (m₁ = m₂)
- For the lines to be **perpendicular**, the product of their slopes must equal -1. (m₁ * m₂ = -1)

### Checking the conditions:
1. **Slopes**:
- m₁ = \(\frac{1}{3}\)
- m₂ = -3

2. **Are they parallel?**:
- \(\frac{1}{3} \neq -3\) (not the same)

3. **Are they perpendicular?**:
- \(\frac{1}{3} \times -3 = -1\) (the product is -1)

Since the slopes are neither equal nor do they satisfy the condition for parallel lines, and they do satisfy the condition for perpendicular lines, we conclude:

**The lines are Perpendicular.**