To determine which linear equation shows a proportional relationship, we need to understand what characterizes a proportional relationship.
A proportional relationship can be described by an equation of the form [tex]\( y = kx \)[/tex], where:
- [tex]\( y \)[/tex] is directly proportional to [tex]\( x \)[/tex].
- [tex]\( k \)[/tex] is a constant, known as the constant of proportionality.
In such an equation, [tex]\( y \)[/tex] is proportional to [tex]\( x \)[/tex] without any additional terms. There are no constants added or subtracted to nor multiplied by the equation aside from [tex]\( k \)[/tex].
Let's examine each given equation to determine if it fits the form [tex]\( y = kx \)[/tex]:
1. [tex]\( y = \frac{1}{7} x - 2 \)[/tex]
- This equation has a term [tex]\(-2\)[/tex] subtracted, so it is not in the form [tex]\( y = kx \)[/tex]. Therefore, it does not describe a proportional relationship.
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]. It describes a proportional relationship because there are no additional terms added or subtracted.
3. [tex]\( y = -7 x + 3 \)[/tex]
- This equation has a term [tex]\( +3 \)[/tex] added, so it is not in the form [tex]\( y = kx \)[/tex]. Therefore, it does not describe a proportional relationship.
4. [tex]\( y = 7 \)[/tex]
- This equation is a constant and independent of [tex]\( x \)[/tex], meaning it does not vary with [tex]\( x \)[/tex]. Therefore, it does not describe a proportional relationship.
Based on this analysis, the correct equation that shows a proportional relationship is:
[tex]\[ y = -\frac{1}{7} x \][/tex]
Thus, the answer is the second equation in the list:
[tex]\[ \boxed{2} \][/tex]
