To determine which linear equation shows a proportional relationship, we need to identify an equation that has the form [tex]\( y = kx \)[/tex], where [tex]\( k \)[/tex] is a constant.
Let's analyze each option step by step:
1. Equation 1: [tex]\( y = \frac{1}{7}x - 2 \)[/tex]
- This equation is in the form of [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- Here, [tex]\( \frac{1}{7} \)[/tex] is the slope and [tex]\(-2\)[/tex] is the y-intercept.
- Since there is a non-zero y-intercept ([tex]\(-2\)[/tex]), it is not a proportional relationship.
2. Equation 2: [tex]\( y = -\frac{1}{7}x \)[/tex]
- This equation is in the form of [tex]\( y = kx \)[/tex] with [tex]\( k = -\frac{1}{7} \)[/tex].
- There is no y-intercept term (i.e., no [tex]\( b \)[/tex] term).
- Since it is in the form [tex]\( y = kx \)[/tex], it is a proportional relationship.
3. Equation 3: [tex]\( y = -7x + 3 \)[/tex]
- This equation is in the form of [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- Here, [tex]\(-7\)[/tex] is the slope and [tex]\(3\)[/tex] is the y-intercept.
- Since there is a non-zero y-intercept ([tex]\(3\)[/tex]), it is not a proportional relationship.
4. Equation 4: [tex]\( y = 7 \)[/tex]
- This is a constant equation where [tex]\( y \)[/tex] is always 7, regardless of the value of [tex]\( x \)[/tex].
- It does not fit the form [tex]\( y = kx \)[/tex] since it does not depend on [tex]\( x \)[/tex].
- Therefore, it is not a proportional relationship.
Based on the analysis, the equation that shows a proportional relationship is:
[tex]\( \boxed{y = -\frac{1}{7}x} \)[/tex]
Thus, the correct answer is option 2.
