To determine which equation represents a proportional relationship with a constant of proportionality equal to [tex]\(\frac{1}{5}\)[/tex], we need to understand that a proportional relationship can be expressed by the equation [tex]\(y = kx\)[/tex], where [tex]\(k\)[/tex] is the constant of proportionality.
Given:
1. [tex]\(y = x - \frac{1}{5}\)[/tex]
2. [tex]\(y = x + \frac{4}{5}\)[/tex]
3. [tex]\(y = \frac{1}{5} x\)[/tex]
4. [tex]\(y = 5 x\)[/tex]
Let's analyze each of these equations to see if they match the form [tex]\(y = kx\)[/tex] and have [tex]\(k = \frac{1}{5}\)[/tex]:
1. [tex]\(y = x - \frac{1}{5}\)[/tex]
This equation is of the form [tex]\(y = mx + b\)[/tex], where [tex]\(m = 1\)[/tex] and [tex]\(b = -\frac{1}{5}\)[/tex]. Here, [tex]\(y\)[/tex] is not directly proportional to [tex]\(x\)[/tex] because of the [tex]\(-\frac{1}{5}\)[/tex] term. Thus, this is not the correct relationship.
2. [tex]\(y = x + \frac{4}{5}\)[/tex]
This equation is also of the form [tex]\(y = mx + b\)[/tex], with [tex]\(m = 1\)[/tex] and [tex]\(b = \frac{4}{5}\)[/tex]. Similarly, [tex]\(y\)[/tex] is not directly proportional to [tex]\(x\)[/tex] because of the [tex]\(\frac{4}{5}\)[/tex] term. So, this is not the correct relationship either.
3. [tex]\(y = \frac{1}{5}x\)[/tex]
This equation is of the form [tex]\(y = kx\)[/tex] where [tex]\(k = \frac{1}{5}\)[/tex]. This perfectly matches the requirement that there be a proportional relationship with a constant of proportionality equal to [tex]\(\frac{1}{5}\)[/tex]. Therefore, this equation represents the correct relationship.
4. [tex]\(y = 5x\)[/tex]
This equation is also of the form [tex]\(y = kx\)[/tex], where [tex]\(k = 5\)[/tex]. This means that the constant of proportionality here is 5, so this does not match the required [tex]\(\frac{1}{5}\)[/tex].
Given the analysis above, the equation that represents a proportional relationship with a constant of proportionality equal to [tex]\(\frac{1}{5}\)[/tex] is:
[tex]\[y = \frac{1}{5} x\][/tex]
Therefore, the correct answer is the third option: [tex]\(3\)[/tex].
