To determine which of the given linear equations shows a proportional relationship, we need to understand what defines a proportional relationship in mathematics.
A proportional relationship is a specific type of linear relationship. It follows the equation:
[tex]\[ y = kx \][/tex]
where [tex]\( k \)[/tex] is the constant of proportionality. This means that in a proportional relationship, the line passes through the origin (0,0) and there is no constant term added or subtracted.
Now, let's analyze each given equation:
1. [tex]\( y = 2x \)[/tex]
2. [tex]\( y = \frac{1}{3}x + 2 \)[/tex]
3. [tex]\( y = -\frac{2}{5}x - 1 \)[/tex]
4. [tex]\( y = 1 \)[/tex]
1. For the equation [tex]\( y = 2x \)[/tex], we can see it is in the form [tex]\( y = kx \)[/tex] (where [tex]\( k = 2 \)[/tex]). There is no additional constant term, so this equation represents a proportional relationship.
2. For the equation [tex]\( y = \frac{1}{3}x + 2 \)[/tex], the form differs because of the added constant term 2. This constant shifts the line to pass through (0,2) instead of the origin, meaning it’s not proportional.
3. For the equation [tex]\( y = -\frac{2}{5}x - 1 \)[/tex], the same situation arises because there is a constant term -1 which shifts the line to pass through (0, -1), indicating that it is not proportional.
4. For the equation [tex]\( y = 1 \)[/tex], this is a horizontal line where [tex]\( y \)[/tex] is always equal to 1 regardless of [tex]\( x \)[/tex]. This doesn’t fit the form [tex]\( y = kx \)[/tex] since it doesn’t involve [tex]\( x \)[/tex] in a linear way other than being zero slope, so it is not a proportional relationship.
Given our analysis, the linear equation that shows a proportional relationship is:
[tex]\[ y = 2x \][/tex]
