Answer :
To determine which linear equation shows a proportional relationship, we need to recall the definition of a proportional relationship. A proportional relationship between two variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex] can be expressed by a linear equation of the form [tex]\( y = kx \)[/tex], where [tex]\( k \)[/tex] is a constant. In other words, the equation should not have any constant term (intercept) other than the coefficient of [tex]\( x \)[/tex].
Let's examine each given equation to see if it fits the form [tex]\( y = kx \)[/tex]:
1. [tex]\( y = -2x - \frac{1}{5} \)[/tex]
- This equation has a constant term [tex]\( -\frac{1}{5} \)[/tex]. Therefore, it is not a proportional relationship.
2. [tex]\( y = \frac{2}{3}x - 1 \)[/tex]
- This equation has a constant term [tex]\( -1 \)[/tex]. Therefore, it is not a proportional relationship.
3. [tex]\( y = 4x + 6 \)[/tex]
- This equation has a constant term [tex]\( 6 \)[/tex]. Therefore, it is not a proportional relationship.
4. [tex]\( y = \frac{3}{4}x \)[/tex]
- This equation has no constant term and fits the form [tex]\( y = kx \)[/tex]. Therefore, it represents a proportional relationship.
Thus, the linear equation that shows a proportional relationship is:
[tex]\[ y = \frac{3}{4}x \][/tex]
So, the correct choice is:
[tex]\[ \boxed{4} \][/tex]
Let's examine each given equation to see if it fits the form [tex]\( y = kx \)[/tex]:
1. [tex]\( y = -2x - \frac{1}{5} \)[/tex]
- This equation has a constant term [tex]\( -\frac{1}{5} \)[/tex]. Therefore, it is not a proportional relationship.
2. [tex]\( y = \frac{2}{3}x - 1 \)[/tex]
- This equation has a constant term [tex]\( -1 \)[/tex]. Therefore, it is not a proportional relationship.
3. [tex]\( y = 4x + 6 \)[/tex]
- This equation has a constant term [tex]\( 6 \)[/tex]. Therefore, it is not a proportional relationship.
4. [tex]\( y = \frac{3}{4}x \)[/tex]
- This equation has no constant term and fits the form [tex]\( y = kx \)[/tex]. Therefore, it represents a proportional relationship.
Thus, the linear equation that shows a proportional relationship is:
[tex]\[ y = \frac{3}{4}x \][/tex]
So, the correct choice is:
[tex]\[ \boxed{4} \][/tex]
