To determine which linear equation shows a proportional relationship, we must first understand what a proportional relationship in terms of a linear equation looks like. A proportional relationship between two variables, [tex]\( x \)[/tex] and [tex]\( y \)[/tex], can be described by a linear equation of the form [tex]\( y = kx \)[/tex], where [tex]\( k \)[/tex] is a constant and there is no added or subtracted constant term.
Let's examine each of the given equations one by one:
1. Equation: [tex]\( y = 2x \)[/tex]
- This equation is in the form [tex]\( y = kx \)[/tex] with [tex]\( k = 2 \)[/tex].
- There are no added or subtracted constants.
- This indicates a proportional relationship.
2. Equation: [tex]\( y = \frac{1}{3}x + 2 \)[/tex]
- This equation has the form [tex]\( y = kx + b \)[/tex] with [tex]\( k = \frac{1}{3} \)[/tex] and [tex]\( b = 2 \)[/tex].
- The added constant term [tex]\( +2 \)[/tex] means this is not a proportional relationship.
3. Equation: [tex]\( y = -\frac{2}{5}x - 1 \)[/tex]
- This equation has the form [tex]\( y = kx + b \)[/tex] with [tex]\( k = -\frac{2}{5} \)[/tex] and [tex]\( b = -1 \)[/tex].
- The added constant term [tex]\( -1 \)[/tex] means this is not a proportional relationship.
4. Equation: [tex]\( y = 1 \)[/tex]
- This is a horizontal line where [tex]\( y \)[/tex] is constantly 1, regardless of [tex]\( x \)[/tex].
- It does not fit the form [tex]\( y = kx \)[/tex] and thus, is not a proportional relationship.
Based on our analysis, the only equation that fits the form [tex]\( y = kx \)[/tex] and hence represents a proportional relationship is [tex]\( y = 2x \)[/tex].
Thus, the linear equation that shows a proportional relationship is:
[tex]\[ \boxed{1} \][/tex]
