Sure! Let's determine the relationships step-by-step.
### Parallel Lines
Lines are parallel if they have the same slope. By examining their equations, we have:
1. \( y = 2x + 4 \)
2. \( 2y = 4x - 7 \implies y = 2x - \frac{7}{2} \)
3. \( 4y = 2x - 4 \implies y = \frac{1}{2}x - 1 \)
4. \( 2y = 4x + 4 \implies y = 2x + 2 \)
5. \( y = -2x + 9 \)
6. \( y = -2x - 2 \)
Comparing the slopes, we see that:
- The slopes of \( y = 2x + 4 \), \( y = 2x - \frac{7}{2} \), and \( y = 2x + 2 \) are all 2.
- The slopes of \( y = -2x + 9 \) and \( y = -2x - 2 \) are both -2.
Therefore, the parallel pairs are:
- \( y = 2x + 4 \) and \( 2y = 4x - 7 \)
- \( y = 2x + 4 \) and \( 2y = 4x + 4 \)
- \( y = -2x + 9 \) and \( y = -2x - 2 \)
### Perpendicular Lines
Lines are perpendicular if the product of their slopes is \(-1\). Examining the slopes again, we have:
- The slopes of \( y = 2x + 4 \), \( y = 2x - \frac{7}{2} \), and \( y = 2x + 2 \) are all 2.
- The slopes of \( y = -2x + 9 \) and \( y = -2x - 2 \) are both -2.
For two lines to be perpendicular,
- The slope of one line multiplied by the slope of the other line should be \(-1\).
However, \( 2 \times -2 = -4 \), which is not equal to \(-1\). Hence, there are no pairs of lines that are perpendicular.
### Neither Parallel Nor Perpendicular
Lines that do not fit either category of being parallel or perpendicular fall into the "neither" group.
Based on the given relationships, the 'neither' category will be for line pairs that do not share the same slope and their slopes' product is not -1.
So,
- The pairs of lines that do not fit into either the parallel or perpendicular categories fall here.
Now, let's place the tiles in their appropriate boxes:
### Parallel
- \( y = 2x + 4 \) and \( 2y = 4x - 7 \)
- \( y = 2x + 4 \) and \( 2y = 4x + 4 \)
- \( y = -2x + 9 \) and \( y = -2x - 2 \)
### Perpendicular
- None
### Neither
- \( y = 2x + 4 \) and \( 4y = 2x - 4 \)
- Any other remaining pairs aside from the ones listed as parallel.
Let's summarize the result in the given format:
### Parallel
- \( y=2x+4 \) and \( 2y=4x-7 \)
- \( y=2x+4 \) and \( 2y=4x+4 \)
- \( y=-2x+9 \) and \( y=-2x-2 \)
### Perpendicular
- None
### Neither
- Remaining line pairs that do not fit into parallel or perpendicular.
I hope this helps!
