Question
Parallel or perpendicular
Y=1/2x+4 and y=2×+10
Y=1/2x+4 and y=2×+10
Answers
GPT-4o mini
To determine whether the lines represented by the equations \( y = \frac{1}{2}x + 4 \) and \( y = 2x + 10 \) are parallel, perpendicular, or neither, we can look at the slopes of the lines.
1. **Identify the slopes:**
- The first equation, \( y = \frac{1}{2}x + 4 \), has a slope \( m_1 = \frac{1}{2} \).
- The second equation, \( y = 2x + 10 \), has a slope \( m_2 = 2 \).
2. **Compare the slopes:**
- Two lines are **parallel** if their slopes are equal: \( m_1 = m_2 \).
- Two lines are **perpendicular** if the product of their slopes is \(-1\): \( m_1 \times m_2 = -1 \).
Now, let's check:
- **Are the lines parallel?**
\[
\frac{1}{2} \neq 2 \quad \text{(not equal, so not parallel)}
\]
- **Are the lines perpendicular?**
\[
m_1 \times m_2 = \frac{1}{2} \times 2 = 1 \quad \text{(not equal to -1, so not perpendicular)}
\]
Since the slopes are neither equal nor do they multiply to \(-1\), the lines represented by the equations \( y = \frac{1}{2}x + 4 \) and \( y = 2x + 10 \) are neither parallel nor perpendicular.
1. **Identify the slopes:**
- The first equation, \( y = \frac{1}{2}x + 4 \), has a slope \( m_1 = \frac{1}{2} \).
- The second equation, \( y = 2x + 10 \), has a slope \( m_2 = 2 \).
2. **Compare the slopes:**
- Two lines are **parallel** if their slopes are equal: \( m_1 = m_2 \).
- Two lines are **perpendicular** if the product of their slopes is \(-1\): \( m_1 \times m_2 = -1 \).
Now, let's check:
- **Are the lines parallel?**
\[
\frac{1}{2} \neq 2 \quad \text{(not equal, so not parallel)}
\]
- **Are the lines perpendicular?**
\[
m_1 \times m_2 = \frac{1}{2} \times 2 = 1 \quad \text{(not equal to -1, so not perpendicular)}
\]
Since the slopes are neither equal nor do they multiply to \(-1\), the lines represented by the equations \( y = \frac{1}{2}x + 4 \) and \( y = 2x + 10 \) are neither parallel nor perpendicular.