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Determine whether or not the equations represent a direct variation. Sort the equations into the appropriate category.

Direct Variation:
[tex]$\square \; y = 3x \; \square$[/tex]
[tex]$\square \; y = (2\pi)x \; \square$[/tex]
[tex]$\square \; -0.5x = y \; \square$[/tex]

Not Direct Variation:
[tex]$\square \; y = 22x + 7 \; \square$[/tex]
[tex]$\square \; y = 4 \; \square$[/tex]


Sagot :

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]
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