Answer :
To determine whether each equation represents a proportional relationship, we need to understand what a proportional relationship means. A proportional relationship between two variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex] can be described by an equation of the form [tex]\( y = kx \)[/tex], where [tex]\( k \)[/tex] is a constant.
Let's analyze each equation to see if it fits this form:
1. Equation: [tex]\( y = 3 + x \)[/tex]
- This equation can be rewritten as [tex]\( y = x + 3 \)[/tex], which is not in the form [tex]\( y = kx \)[/tex] because of the additional constant 3.
- Conclusion: Not a Proportional Relationship
2. Equation: [tex]\( y = 1.6x \)[/tex]
- This equation is already in the form [tex]\( y = kx \)[/tex], with [tex]\( k = 1.6 \)[/tex].
- Conclusion: Proportional Relationship
3. Equation: [tex]\( y = \frac{3}{4x} \)[/tex]
- This equation can be written as [tex]\( y = \frac{3}{4} \cdot \frac{1}{x} \)[/tex], which is not in the form [tex]\( y = kx \)[/tex] because of the division by [tex]\( x \)[/tex].
- Conclusion: Not a Proportional Relationship
4. Equation: [tex]\( y = x \)[/tex]
- This equation is already in the form [tex]\( y = kx \)[/tex], with [tex]\( k = 1 \)[/tex].
- Conclusion: Proportional Relationship
5. Equation: [tex]\( y = 12x \)[/tex]
- This equation is already in the form [tex]\( y = kx \)[/tex], with [tex]\( k = 12 \)[/tex].
- Conclusion: Proportional Relationship
6. Equation: [tex]\( y = 2x + 1 \)[/tex]
- This equation can be rewritten as [tex]\( y = 2x + 1 \)[/tex], which is not in the form [tex]\( y = kx \)[/tex] because of the additional constant 1.
- Conclusion: Not a Proportional Relationship
Summarizing, we have:
- [tex]\( y = 3 + x \)[/tex]: Not a Proportional Relationship
- [tex]\( y = 1.6x \)[/tex]: Proportional Relationship
- [tex]\( y = \frac{3}{4x} \)[/tex]: Not a Proportional Relationship
- [tex]\( y = x \)[/tex]: Proportional Relationship
- [tex]\( y = 12x \)[/tex]: Proportional Relationship
- [tex]\( y = 2x + 1 \)[/tex]: Not a Proportional Relationship
Let's analyze each equation to see if it fits this form:
1. Equation: [tex]\( y = 3 + x \)[/tex]
- This equation can be rewritten as [tex]\( y = x + 3 \)[/tex], which is not in the form [tex]\( y = kx \)[/tex] because of the additional constant 3.
- Conclusion: Not a Proportional Relationship
2. Equation: [tex]\( y = 1.6x \)[/tex]
- This equation is already in the form [tex]\( y = kx \)[/tex], with [tex]\( k = 1.6 \)[/tex].
- Conclusion: Proportional Relationship
3. Equation: [tex]\( y = \frac{3}{4x} \)[/tex]
- This equation can be written as [tex]\( y = \frac{3}{4} \cdot \frac{1}{x} \)[/tex], which is not in the form [tex]\( y = kx \)[/tex] because of the division by [tex]\( x \)[/tex].
- Conclusion: Not a Proportional Relationship
4. Equation: [tex]\( y = x \)[/tex]
- This equation is already in the form [tex]\( y = kx \)[/tex], with [tex]\( k = 1 \)[/tex].
- Conclusion: Proportional Relationship
5. Equation: [tex]\( y = 12x \)[/tex]
- This equation is already in the form [tex]\( y = kx \)[/tex], with [tex]\( k = 12 \)[/tex].
- Conclusion: Proportional Relationship
6. Equation: [tex]\( y = 2x + 1 \)[/tex]
- This equation can be rewritten as [tex]\( y = 2x + 1 \)[/tex], which is not in the form [tex]\( y = kx \)[/tex] because of the additional constant 1.
- Conclusion: Not a Proportional Relationship
Summarizing, we have:
- [tex]\( y = 3 + x \)[/tex]: Not a Proportional Relationship
- [tex]\( y = 1.6x \)[/tex]: Proportional Relationship
- [tex]\( y = \frac{3}{4x} \)[/tex]: Not a Proportional Relationship
- [tex]\( y = x \)[/tex]: Proportional Relationship
- [tex]\( y = 12x \)[/tex]: Proportional Relationship
- [tex]\( y = 2x + 1 \)[/tex]: Not a Proportional Relationship
