To determine which linear equation shows a proportional relationship, we need to understand that a proportional relationship is one where the equation takes the form [tex]\( y = kx \)[/tex], where [tex]\( k \)[/tex] is a constant and there is no constant term added or subtracted (no y-intercept other than zero).
Let's analyze each option step-by-step to see if it fits this form:
1. [tex]\( y = 2x + 5 \)[/tex]
- This equation has the format [tex]\( y = kx + c \)[/tex] with [tex]\( k = 2 \)[/tex] and [tex]\( c = 5 \)[/tex]. The presence of the constant term [tex]\( +5 \)[/tex] (non-zero y-intercept) indicates that it is not a proportional relationship.
2. [tex]\( y = \frac{1}{5}x - 7 \)[/tex]
- This equation is of the form [tex]\( y = kx + c \)[/tex] with [tex]\( k = \frac{1}{5} \)[/tex] and [tex]\( c = -7 \)[/tex]. The presence of the constant term [tex]\( -7 \)[/tex] (non-zero y-intercept) indicates that it is not a proportional relationship.
3. [tex]\( y = -\frac{1}{5}x \)[/tex]
- This equation fits the form [tex]\( y = kx \)[/tex], where [tex]\( k = -\frac{1}{5} \)[/tex]. There is no constant term added or subtracted (no non-zero y-intercept), indicating that this is a proportional relationship.
4. [tex]\( y = -5 \)[/tex]
- This equation describes a constant function where [tex]\( y \)[/tex] is always [tex]\(-5\)[/tex], regardless of [tex]\( x \)[/tex]. Since [tex]\( y \)[/tex] does not vary with [tex]\( x \)[/tex], this cannot be considered a proportional relationship.
Therefore, the linear equation that shows a proportional relationship is:
[tex]\[ y = -\frac{1}{5}x \][/tex]
So, the correct answer is option 3.
