Sure, let's analyze each of the given lines and then determine their relationships—in terms of being parallel, perpendicular, or neither—step by step.
### Line 1
The equation for Line 1 is:
[tex]\[ 5y = 3x + 6 \][/tex]
First, let's turn this into slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y = \frac{3}{5}x + \frac{6}{5} \][/tex]
The slope [tex]\( m_1 \)[/tex] for Line 1 is:
[tex]\[ m_1 = \frac{3}{5} \][/tex]
### Line 2
The equation for Line 2 is:
[tex]\[ y = \frac{3}{5}x - 5 \][/tex]
This is already in slope-intercept form [tex]\( y = mx + b \)[/tex].
The slope [tex]\( m_2 \)[/tex] for Line 2 is:
[tex]\[ m_2 = \frac{3}{5} \][/tex]
### Line 3
The equation for Line 3 is:
[tex]\[ 10x - 6y = -8 \][/tex]
First, let's convert this into slope-intercept form [tex]\( y = mx + b \)[/tex].
Starting by isolating [tex]\( y \)[/tex]:
[tex]\[ -6y = -10x - 8 \][/tex]
Divide everything by [tex]\(-6\)[/tex]:
[tex]\[ y = \frac{10}{6}x + \frac{8}{6} \][/tex]
Simplify the fractions:
[tex]\[ y = \frac{5}{3}x + \frac{4}{3} \][/tex]
The slope [tex]\( m_3 \)[/tex] for Line 3 is:
[tex]\[ m_3 = \frac{5}{3} \][/tex]
### Comparing the Slopes of the Lines
1. Line 1 and Line 2:
- Slope of Line 1: [tex]\( m_1 = \frac{3}{5} \)[/tex]
- Slope of Line 2: [tex]\( m_2 = \frac{3}{5} \)[/tex]
Since [tex]\( m_1 = m_2 \)[/tex], Line 1 and Line 2 are parallel.
2. Line 1 and Line 3:
- Slope of Line 1: [tex]\( m_1 = \frac{3}{5} \)[/tex]
- Slope of Line 3: [tex]\( m_3 = \frac{5}{3} \)[/tex]
To check if the lines are perpendicular, we check the product of their slopes:
[tex]\[ m_1 \cdot m_3 = \frac{3}{5} \cdot \frac{5}{3} = 1 \neq -1 \][/tex]
Since [tex]\( m_1 \neq m_3 \)[/tex] and [tex]\( m_1 \cdot m_3 \neq -1 \)[/tex], Line 1 and Line 3 are neither parallel nor perpendicular.
3. Line 2 and Line 3:
- Slope of Line 2: [tex]\( m_2 = \frac{3}{5} \)[/tex]
- Slope of Line 3: [tex]\( m_3 = \frac{5}{3} \)[/tex]
To check again if the lines are perpendicular:
[tex]\[ m_2 \cdot m_3 = \frac{3}{5} \cdot \frac{5}{3} = 1 \neq -1 \][/tex]
Since [tex]\( m_2 \neq m_3 \)[/tex] and [tex]\( m_2 \cdot m_3 \neq -1 \)[/tex], Line 2 and Line 3 are neither parallel nor perpendicular.
### Conclusion
- Line 1 and Line 2 are parallel.
- Line 1 and Line 3 are neither parallel nor perpendicular.
- Line 2 and Line 3 are neither parallel nor perpendicular.
