To solve this problem, we need to determine the slopes of each pair of lines and then compare them. Recall that:
- Slopes of parallel lines are equal.
- Slopes of perpendicular lines are negative reciprocals of each other.
Given equations:
1. [tex]\( y = 2x + 4 \)[/tex]
2. [tex]\( 2y = 4x + 4 \)[/tex]
3. [tex]\( 4y = 2x - 4 \)[/tex]
4. [tex]\( 2y = 4x - 7 \)[/tex]
5. [tex]\( y = -2x - 2 \)[/tex]
6. [tex]\( y = -2x + 9 \)[/tex]
First, simplify each equation to the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope.
1. [tex]\( y = 2x + 4 \)[/tex]
- Slope [tex]\( m = 2 \)[/tex]
2. [tex]\( 2y = 4x + 4 \)[/tex]
- Divide both sides by 2: [tex]\( y = 2x + 2 \)[/tex]
- Slope [tex]\( m = 2 \)[/tex]
3. [tex]\( 4y = 2x - 4 \)[/tex]
- Divide both sides by 4: [tex]\( y = \frac{1}{2}x - 1 \)[/tex]
- Slope [tex]\( m = \frac{1}{2} \)[/tex]
4. [tex]\( 2y = 4x - 7 \)[/tex]
- Divide both sides by 2: [tex]\( y = 2x - \frac{7}{2} \)[/tex]
- Slope [tex]\( m = 2 \)[/tex]
5. [tex]\( y = -2x - 2 \)[/tex]
- Slope [tex]\( m = -2 \)[/tex]
6. [tex]\( y = -2x + 9 \)[/tex]
- Slope [tex]\( m = -2 \)[/tex]
Next, compare the slopes:
- Lines 1, 2, and 4 all have the slope [tex]\( m = 2 \)[/tex], making them parallel to each other.
- Lines 5 and 6 both have the slope [tex]\( m = -2 \)[/tex], making them parallel to each other.
- No lines have slopes that are negative reciprocals, so none are perpendicular.
- Lines 3 and any of the other lines (1, 2, 4, 5, 6) have different slopes, so they are neither parallel nor perpendicular to each other.
Thus:
Neither:
[tex]\[ \boxed{4y = 2x - 4 \text{ and any of the other lines (e.g., line 1)}} \][/tex]
Parallel:
[tex]\[ \text{Box 1: } y = 2x + 4 \text{ and } 2y = 4x + 4 \][/tex]
[tex]\[ \text{Box 2: } y = -2x - 2 \text{ and } y = -2x + 9 \][/tex]
No lines are perpendicular in the given set.
