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Sagot :
Let's work step-by-step to determine the relationships between each pair of given lines based on their slopes.
1. Pair: [tex]\( y=2x+4 \)[/tex] and [tex]\( 4y=2x-4 \)[/tex]
- Rewriting [tex]\( 4y=2x-4 \)[/tex] in slope-intercept form:
[tex]\[ 4y = 2x - 4 \implies y = \frac{1}{2}x - 1 \][/tex]
- The slopes are 2 and [tex]\(\frac{1}{2}\)[/tex]. The slopes are not equal and their product is not [tex]\(-1\)[/tex], so the lines are neither parallel nor perpendicular.
2. Pair: [tex]\( 4y=2x-4 \)[/tex] and [tex]\( 2y=4x-7 \)[/tex]
- Rewriting [tex]\( 2y=4x-7 \)[/tex]:
[tex]\[ 2y = 4x - 7 \implies y = 2x - \frac{7}{2} \][/tex]
- The slopes are [tex]\(\frac{1}{2}\)[/tex] and 2. The slopes are not equal and their product is not [tex]\(-1\)[/tex], so the lines are neither parallel nor perpendicular.
3. Pair: [tex]\( 2y=4x-7 \)[/tex] and [tex]\( y=-2x+9 \)[/tex]
- The slopes are 2 and -2. The slopes are not equal and their product is not [tex]\(-1\)[/tex], so the lines are neither parallel nor perpendicular.
4. Pair: [tex]\( y=-2x+9 \)[/tex] and [tex]\( y=-2x-2 \)[/tex]
- The slopes are both -2. The slopes are equal, so the lines are parallel.
5. Pair: [tex]\( y=-2x-2 \)[/tex] and [tex]\( 2y=4x+4 \)[/tex]
- Rewriting [tex]\( 2y=4x+4 \)[/tex] in slope-intercept form:
[tex]\[ 2y = 4x + 4 \implies y = 2x + 2 \][/tex]
- The slopes are -2 and 2. The slopes are not equal and their product is not [tex]\(-1\)[/tex], so the lines are neither parallel nor perpendicular.
6. Pair: [tex]\( 2y=4x+4 \)[/tex] and [tex]\( y=2x+4 \)[/tex]
- The slopes are both 2. The slopes are equal, so the lines are parallel.
Now, let's organize the pairs based on the determined relationships:
### Neither
- [tex]\( y=2x+4 \)[/tex] and [tex]\( 4y=2x-4 \)[/tex]
- [tex]\( 4y=2x-4 \)[/tex] and [tex]\( 2y=4x-7 \)[/tex]
- [tex]\( 2y=4x-7 \)[/tex] and [tex]\( y=-2x+9 \)[/tex]
- [tex]\( y=-2x-2 \)[/tex] and [tex]\( 2y=4x+4 \)[/tex]
### Parallel
- [tex]\( y=-2x+9 \)[/tex] and [tex]\( y=-2x-2 \)[/tex]
- [tex]\( 2y=4x+4 \)[/tex] and [tex]\( y=2x+4 \)[/tex]
Therefore, the pairs are arranged as follows:
Neither
- [tex]\( y=2x+4 \)[/tex] and [tex]\( 4y=2x-4 \)[/tex]
- [tex]\( 4y=2x-4 \)[/tex] and [tex]\( 2y=4x-7 \)[/tex]
- [tex]\( 2y=4x-7 \)[/tex] and [tex]\( y=-2x+9 \)[/tex]
- [tex]\( y=-2x-2 \)[/tex] and [tex]\( 2y=4x+4 \)[/tex]
Parallel
- [tex]\( y=-2x+9 \)[/tex] and [tex]\( y=-2x-2 \)[/tex]
- [tex]\( 2y=4x+4 \)[/tex] and [tex]\( y=2x+4 \)[/tex]
1. Pair: [tex]\( y=2x+4 \)[/tex] and [tex]\( 4y=2x-4 \)[/tex]
- Rewriting [tex]\( 4y=2x-4 \)[/tex] in slope-intercept form:
[tex]\[ 4y = 2x - 4 \implies y = \frac{1}{2}x - 1 \][/tex]
- The slopes are 2 and [tex]\(\frac{1}{2}\)[/tex]. The slopes are not equal and their product is not [tex]\(-1\)[/tex], so the lines are neither parallel nor perpendicular.
2. Pair: [tex]\( 4y=2x-4 \)[/tex] and [tex]\( 2y=4x-7 \)[/tex]
- Rewriting [tex]\( 2y=4x-7 \)[/tex]:
[tex]\[ 2y = 4x - 7 \implies y = 2x - \frac{7}{2} \][/tex]
- The slopes are [tex]\(\frac{1}{2}\)[/tex] and 2. The slopes are not equal and their product is not [tex]\(-1\)[/tex], so the lines are neither parallel nor perpendicular.
3. Pair: [tex]\( 2y=4x-7 \)[/tex] and [tex]\( y=-2x+9 \)[/tex]
- The slopes are 2 and -2. The slopes are not equal and their product is not [tex]\(-1\)[/tex], so the lines are neither parallel nor perpendicular.
4. Pair: [tex]\( y=-2x+9 \)[/tex] and [tex]\( y=-2x-2 \)[/tex]
- The slopes are both -2. The slopes are equal, so the lines are parallel.
5. Pair: [tex]\( y=-2x-2 \)[/tex] and [tex]\( 2y=4x+4 \)[/tex]
- Rewriting [tex]\( 2y=4x+4 \)[/tex] in slope-intercept form:
[tex]\[ 2y = 4x + 4 \implies y = 2x + 2 \][/tex]
- The slopes are -2 and 2. The slopes are not equal and their product is not [tex]\(-1\)[/tex], so the lines are neither parallel nor perpendicular.
6. Pair: [tex]\( 2y=4x+4 \)[/tex] and [tex]\( y=2x+4 \)[/tex]
- The slopes are both 2. The slopes are equal, so the lines are parallel.
Now, let's organize the pairs based on the determined relationships:
### Neither
- [tex]\( y=2x+4 \)[/tex] and [tex]\( 4y=2x-4 \)[/tex]
- [tex]\( 4y=2x-4 \)[/tex] and [tex]\( 2y=4x-7 \)[/tex]
- [tex]\( 2y=4x-7 \)[/tex] and [tex]\( y=-2x+9 \)[/tex]
- [tex]\( y=-2x-2 \)[/tex] and [tex]\( 2y=4x+4 \)[/tex]
### Parallel
- [tex]\( y=-2x+9 \)[/tex] and [tex]\( y=-2x-2 \)[/tex]
- [tex]\( 2y=4x+4 \)[/tex] and [tex]\( y=2x+4 \)[/tex]
Therefore, the pairs are arranged as follows:
Neither
- [tex]\( y=2x+4 \)[/tex] and [tex]\( 4y=2x-4 \)[/tex]
- [tex]\( 4y=2x-4 \)[/tex] and [tex]\( 2y=4x-7 \)[/tex]
- [tex]\( 2y=4x-7 \)[/tex] and [tex]\( y=-2x+9 \)[/tex]
- [tex]\( y=-2x-2 \)[/tex] and [tex]\( 2y=4x+4 \)[/tex]
Parallel
- [tex]\( y=-2x+9 \)[/tex] and [tex]\( y=-2x-2 \)[/tex]
- [tex]\( 2y=4x+4 \)[/tex] and [tex]\( y=2x+4 \)[/tex]
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