To determine whether the given equations represent a direct variation, we need to understand what direct variation means. Direct variation implies a relationship of the form [tex]\( y = kx \)[/tex] or [tex]\( x = ky \)[/tex], where [tex]\( k \)[/tex] is a constant.
Let's analyze each given equation step-by-step:
1. [tex]\( x = -1 \)[/tex]:
- This equation states that [tex]\( x \)[/tex] is always equal to [tex]\(-1\)[/tex], regardless of [tex]\( y \)[/tex]. This is not in the form [tex]\( y = kx \)[/tex] or [tex]\( x = ky \)[/tex], so it does not represent a direct variation.
2. [tex]\( y = \left(\frac{2}{7}\right) x \)[/tex]:
- This equation is of the form [tex]\( y = kx \)[/tex], where [tex]\( k = \frac{2}{7} \)[/tex]. This is a direct variation with [tex]\( k = \frac{2}{7} \)[/tex].
3. [tex]\( -0.5 x = y \)[/tex]:
- This equation can be rewritten as [tex]\( y = -0.5x \)[/tex], which is of the form [tex]\( y = kx \)[/tex] with [tex]\( k = -0.5 \)[/tex]. Therefore, this is a direct variation with [tex]\( k = -0.5 \)[/tex].
4. [tex]\( y = 2.2 x + 7 \)[/tex]:
- This equation includes an additional constant [tex]\(+7\)[/tex]. Therefore, it cannot be written in the form [tex]\( y = kx \)[/tex] because of the additional term. This is not a direct variation.
5. [tex]\( y = 4 \)[/tex]:
- This equation states that [tex]\( y \)[/tex] is always equal to 4, regardless of [tex]\( x \)[/tex]. This does not fit the form [tex]\( y = kx \)[/tex] or [tex]\( x = ky \)[/tex], so it is not a direct variation.
Now, I'll categorize the equations based on whether they represent direct variation or not.
Direct Variation:
- [tex]\( y = \left(\frac{2}{7}\right) x \)[/tex]
- [tex]\( -0.5 x = y \)[/tex]
Not Direct Variation:
- [tex]\( x = -1 \)[/tex]
- [tex]\( y = 2.2 x + 7 \)[/tex]
- [tex]\( y = 4 \)[/tex]
So, the complete sorted categories are:
- Direct Variation: [tex]\( y = \left(\frac{2}{7}\right) x \)[/tex], [tex]\( -0.5 x = y \)[/tex]
- Not Direct Variation: [tex]\( x = -1 \)[/tex], [tex]\( y = 2.2 x + 7 \)[/tex], [tex]\( y = 4 \)[/tex]
