To determine which linear equation shows a proportional relationship, we need to look for an equation of the form [tex]\( y = kx \)[/tex], where [tex]\( k \)[/tex] is a constant and there is no other term added or subtracted.
Let's analyze each given option one by one:
1. [tex]\( y = \frac{1}{7}x - 2 \)[/tex]
- This equation includes an additional constant term (-2), which means it is not a purely proportional relationship. A proportional relationship doesn't have an extra constant term added or subtracted.
2. [tex]\( y = -\frac{1}{7}x \)[/tex]
- This equation is in the form [tex]\( y = kx \)[/tex], where [tex]\( k = -\frac{1}{7} \)[/tex]. There are no additional constant terms. This equation represents a proportional relationship.
3. [tex]\( y = -7x + 3 \)[/tex]
- Similar to the first option, this equation includes an additional constant term (+3), so it is not a purely proportional relationship.
4. [tex]\( y = 7 \)[/tex]
- In this equation, [tex]\( y \)[/tex] does not depend on [tex]\( x \)[/tex]. It represents a constant value and does not show a proportional relationship.
Given this analysis, the correct linear equation that shows a proportional relationship is:
[tex]\[ y = -\frac{1}{7}x \][/tex]
Therefore, the correct option is the second one.
