To determine which equations represent a direct variation, we need to check if they can be written in the form [tex]\( y = kx \)[/tex], where [tex]\( k \)[/tex] is a constant. Let's analyze each equation step by step:
1. Equation: [tex]\( y = 3x \)[/tex]
- This equation is already in the form [tex]\( y = kx \)[/tex], where [tex]\( k = 3 \)[/tex].
- Therefore, [tex]\( y = 3x \)[/tex] is a direct variation.
2. Equation: [tex]\( x = -1 \)[/tex]
- This equation represents a vertical line and does not have the form [tex]\( y = kx \)[/tex].
- Therefore, [tex]\( x = -1 \)[/tex] is not a direct variation.
3. Equation: [tex]\( y = \frac{2}{7}x \)[/tex]
- This equation is already in the form [tex]\( y = kx \)[/tex], where [tex]\( k = \frac{2}{7} \)[/tex].
- Therefore, [tex]\( y = \frac{2}{7}x \)[/tex] is a direct variation.
4. Equation: [tex]\( -0.5x = y \)[/tex]
- This equation can be rearranged to [tex]\( y = -0.5x \)[/tex], which is in the form [tex]\( y = kx \)[/tex] with [tex]\( k = -0.5 \)[/tex].
- Therefore, [tex]\( -0.5x = y \)[/tex] is a direct variation.
5. Equation: [tex]\( y = 2.2x + 7 \)[/tex]
- This equation has a constant term (+7) added, which prevents it from being in the form [tex]\( y = kx \)[/tex].
- Therefore, [tex]\( y = 2.2x + 7 \)[/tex] is not a direct variation.
6. Equation: [tex]\( y = 4 \)[/tex]
- This equation represents a horizontal line where [tex]\( y \)[/tex] is constant and does not change with [tex]\( x \)[/tex].
- It does not have the form [tex]\( y = kx \)[/tex].
- Therefore, [tex]\( y = 4 \)[/tex] is not a direct variation.
Based on this analysis, we can sort the equations as follows:
Direct Variation:
- [tex]\( y = 3x \)[/tex]
- [tex]\( y = \frac{2}{7}x \)[/tex]
- [tex]\( -0.5x = y \)[/tex]
Not Direct Variation:
- [tex]\( x = -1 \)[/tex]
- [tex]\( y = 2.2x + 7 \)[/tex]
- [tex]\( y = 4 \)[/tex]
