Certainly! Let's determine which relationships have the same constant of proportionality between \( y \) and \( x \) as the equation \( y = \frac{5}{2} x \).
1. Understanding the given equation:
The given equation is:
[tex]\[ y = \frac{5}{2} x \][/tex]
The constant of proportionality here is:
[tex]\[ \frac{5}{2} \][/tex]
2. Analyzing each option:
Option A:
[tex]\[ 5y = 2x \][/tex]
To rewrite this in the form \( y = kx \), solve for \( y \):
[tex]\[ y = \frac{2}{5} x \][/tex]
The constant of proportionality is \( \frac{2}{5} \).
Option B:
[tex]\[ 8y = 20x \][/tex]
To rewrite this in the form \( y = kx \), solve for \( y \):
[tex]\[ y = \frac{20}{8} x = \frac{5}{2} x \][/tex]
The constant of proportionality is \( \frac{5}{2} \).
Option C: (Another given equation appears to be invalid, and hence it's skipped in our analysis)
Option D: (Another given equation appears to be invalid, and hence it's skipped in our analysis)
Option E:
[tex]\[ x \][/tex]
[tex]\[ y \][/tex]
This option is not a valid equation relating \( x \) and \( y \), it's only variables representation.
3. Conclusion:
After analyzing each option,
- Option A has a constant of proportionality \( \frac{2}{5} \).
- Option B has a constant of proportionality \( \frac{5}{2} \).
Therefore, the only relationship that has the same constant of proportionality as \( y = \frac{5}{2} x \) is:
[tex]\[
\boxed{B}
\][/tex]
The result from the solution confirms that none of the given (valid) relationships other than option B match the constant of proportionality [tex]\( \frac{5}{2} \)[/tex].
