Sure, let's determine whether each given equation represents a direct variation or not. A direct variation equation can be written in the form [tex]\( y = kx \)[/tex], where [tex]\( k \)[/tex] is a constant.
1. Analyzing [tex]\( y = 3x \)[/tex]:
- This equation is in the form [tex]\( y = kx \)[/tex] where [tex]\( k = 3 \)[/tex].
- Therefore, [tex]\( y = 3x \)[/tex] represents a direct variation.
2. Analyzing [tex]\( y = (2\pi)x \)[/tex]:
- Although this equation looks like it’s in the form [tex]\( y = kx \)[/tex], it isn't because it transforms the constant [tex]\( k \)[/tex]. However, the unconventional form should be carefully checked.
- Constant [tex]\( 2\pi \)[/tex] is a multiplication term here so it represents a direct variation as well.
3. Analyzing [tex]\( -0.5x = y \)[/tex]:
- Rewriting the equation as [tex]\( y = -0.5x \)[/tex] shows that it is in the form [tex]\( y = kx \)[/tex] where [tex]\( k = -0.5 \)[/tex].
- Therefore, [tex]\( -0.5x = y \)[/tex] represents a direct variation.
4. Analyzing [tex]\( y = 22x + 7 \)[/tex]:
- This equation is not in the form [tex]\( y = kx \)[/tex] because of the additional "+7" term.
- Therefore, [tex]\( y = 22x + 7 \)[/tex] does not represent a direct variation.
5. Analyzing [tex]\( y = 4 \)[/tex]:
- This equation represents a horizontal line and does not fit the form [tex]\( y = kx \)[/tex] for any non-zero [tex]\( k \)[/tex].
- Therefore, [tex]\( y = 4 \)[/tex] does not represent a direct variation.
Now, let's categorize them:
#### Represent Direct Variation:
- [tex]\( y=3x \)[/tex]
- [tex]\( y=(2\pi)x \)[/tex]
- [tex]\( -0.5x=y \)[/tex]
#### Do Not Represent Direct Variation:
- [tex]\( y=22x+7 \)[/tex]
- [tex]\( y=4 \)[/tex]
Therefore, the final sorted list is:
Direct Variation:
- [tex]$y=3 x$[/tex]
Not Direct Variation:
- [tex]$y=(2 \pi) x$[/tex]
- [tex]$-0.5 x=y$[/tex]
- [tex]$y=22 x+7$[/tex]
- [tex]$y=4$[/tex]
