To determine which equation represents a proportional relationship with a constant of proportionality equal to 2, we need to evaluate each given equation and identify which one adheres to the definition of a proportional relationship.
A proportional relationship between two variables [tex]\( y \)[/tex] and [tex]\( x \)[/tex] can be expressed as:
[tex]\[ y = kx \][/tex]
where [tex]\( k \)[/tex] is the constant of proportionality. For this problem, we are given that the constant of proportionality, [tex]\( k \)[/tex], should be 2. Thus, we look for an equation of the form:
[tex]\[ y = 2x \][/tex]
Now, let's examine each of the given equations:
1. Equation: [tex]\( y = x + 2 \)[/tex]
- This is a linear equation, but it is not proportional because it has an additional constant term (the +2). In a proportional relationship, the graph must pass through the origin (0,0), and this equation does not satisfy that condition.
2. Equation: [tex]\( y = \frac{x}{2} \)[/tex]
- This equation represents a proportional relationship with a constant of proportionality of [tex]\( \frac{1}{2} \)[/tex]. Since our required constant is 2, this equation does not satisfy the condition.
3. Equation: [tex]\( y = 2x \)[/tex]
- This equation represents a proportional relationship where [tex]\( k = 2 \)[/tex]. The relationship is directly proportional, and the graph passes through the origin. This matches our requirement perfectly.
4. Equation: [tex]\( y = 2 \)[/tex]
- This is a constant function, not a relationship between [tex]\( y \)[/tex] and [tex]\( x \)[/tex] that changes with [tex]\( x \)[/tex]. It does not represent a proportional relationship.
Therefore, the correct equation that represents a proportional relationship with a constant of proportionality equal to 2 is:
[tex]\[ y = 2x \][/tex]
The index of the correct equation, given the provided options, is:
[tex]\[ \boxed{3} \][/tex]
