To determine which equation has a proportional relationship, we need to identify which equations can be expressed in the form \( y = kx \), where \( k \) is a constant. In this form, \( y \) is directly proportional to \( x \), meaning that the ratio \( \frac{y}{x} \) is constant for all values of \( x \) (excluding \( x = 0 \)).
Let's analyze each equation:
A. \( y = 4x + 2 \)
This equation is not proportional because of the constant term \( +2 \). It can be rewritten as \( y = 4x + 2 \) and doesn't fit the form \( y = kx \).
B. \( y = x + 3 \)
This equation is not proportional because of the constant term \( +3 \). It doesn't fit the form \( y = kx \).
C. \( y = 8x \)
This equation is proportional because it can be expressed in the form \( y = kx \) where \( k = 8 \).
D. \( y = 2x + 1 \)
This equation is not proportional because of the constant term \( +1 \). It doesn't fit the form \( y = kx \).
In summary, the equation with a proportional relationship is:
C. \( y = 8x \)