To determine which line has a constant of proportionality between \( y \) and \( x \) of \(\frac{5}{4}\), we need to understand the relationship between \( y \) and \( x \).
The proportional relationship can be expressed as:
[tex]\[ y = kx \][/tex]
where \( k \) is the constant of proportionality.
Given that the constant of proportionality is \(\frac{5}{4}\), we can substitute \( k \) with \(\frac{5}{4}\). Therefore, the equation of the line becomes:
[tex]\[ y = \frac{5}{4}x \][/tex]
To verify, let's substitute and solve for \( y \) when \( x \) is an arbitrary number, say \( x = 4 \):
[tex]\[ y = \frac{5}{4} \times 4 \][/tex]
[tex]\[ y = 5 \][/tex]
This simple calculation shows that for every value of \( x \), \( y \) will be \(\frac{5}{4}\) times that value.
Therefore, the line that has a constant of proportionality between \( y \) and \( x \) of \(\frac{5}{4}\) is:
[tex]\[ y = \frac{5}{4}x \][/tex]
This is the detailed, step-by-step process to identify such a line.
