To determine which linear equation shows a proportional relationship, we first need to understand what constitutes a proportional relationship in mathematics. A proportional relationship between two variables can be described by an equation of the form [tex]\( y = kx \)[/tex], where [tex]\( k \)[/tex] is a constant and there is no additional constant term. This type of equation demonstrates that [tex]\( y \)[/tex] changes directly with [tex]\( x \)[/tex] and that the ratio [tex]\( \frac{y}{x} \)[/tex] is always [tex]\( k \)[/tex].
Let’s analyze each of the given equations:
1. [tex]\( y = -2 \)[/tex]:
- This equation implies that [tex]\( y \)[/tex] is always -2, regardless of [tex]\( x \)[/tex]. This is a constant function, not a proportional relationship. No matter what changes occur in [tex]\( x \)[/tex], [tex]\( y \)[/tex] does not change proportionally.
2. [tex]\( y = 3x + 1 \)[/tex]:
- Here, [tex]\( y \)[/tex] depends on [tex]\( x \)[/tex] but there is an additional constant term (+1). The presence of this constant term means that the ratio [tex]\( \frac{y}{x} \)[/tex] is not consistent and changes with different values of [tex]\( x \)[/tex]. Therefore, this is not a proportional relationship.
3. [tex]\( y = \frac{3}{2}x \)[/tex]:
- In this equation, [tex]\( y \)[/tex] is directly proportional to [tex]\( x \)[/tex] with the constant of proportionality being [tex]\( \frac{3}{2} \)[/tex]. This fits the form [tex]\( y = kx \)[/tex] perfectly without any additional terms. Hence, this shows a proportional relationship.
4. [tex]\( y = \frac{1}{2}x - 3 \)[/tex]:
- This equation has an additional constant term (-3). Because of this constant term, [tex]\( y \)[/tex] does not change proportionally with [tex]\( x \)[/tex]. Thus, this is not a proportional relationship.
By analyzing the above equations, we can conclude that the equation which shows a proportional relationship is:
[tex]\[ y = \frac{3}{2}x \][/tex]
Therefore, the correct answer is equation 3.
