To determine which linear equation shows a proportional relationship, we need to understand what a proportional relationship is in the context of linear equations.
A proportional relationship between two variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex] can be expressed in the form:
[tex]\[ y = kx \][/tex]
where [tex]\( k \)[/tex] is a constant. The key characteristics of a proportional relationship are:
1. The equation is linear.
2. The line passes through the origin [tex]\((0,0)\)[/tex].
3. There is no constant term added or subtracted (i.e., the equation does not include a y-intercept term other than zero).
Let's analyze each given equation to see which one fits this form:
1. Equation 1: [tex]\( y = \frac{1}{7}x - 2 \)[/tex]
- This equation is in the form [tex]\( y = mx + c \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( c \)[/tex] is the y-intercept.
- The y-intercept [tex]\( c \)[/tex] here is [tex]\(-2\)[/tex].
- Since there is a y-intercept that is not zero, this equation does not represent a proportional relationship.
2. Equation 2: [tex]\( y = -\frac{1}{7}x \)[/tex]
- This equation is in the form [tex]\( y = kx \)[/tex], where [tex]\( k = -\frac{1}{7} \)[/tex].
- There is no additional constant term; hence, the y-intercept is zero.
- Therefore, this is a proportional relationship as it fits the [tex]\( y = kx \)[/tex] form perfectly and passes through the origin.
3. Equation 3: [tex]\( y = -7x + 3 \)[/tex]
- This equation is in the form [tex]\( y = mx + c \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( c \)[/tex] is the y-intercept.
- The y-intercept [tex]\( c \)[/tex] here is [tex]\( 3 \)[/tex].
- Since there is a y-intercept that is not zero, this equation does not represent a proportional relationship.
4. Equation 4: [tex]\( y = 7 \)[/tex]
- This equation represents a horizontal line where the value of [tex]\( y \)[/tex] is always [tex]\( 7 \)[/tex] regardless of [tex]\( x \)[/tex].
- It does not pass through the origin and is not in the form [tex]\( y = kx \)[/tex].
- Thus, this equation does not represent a proportional relationship.
After analyzing all the equations, we conclude that:
The linear equation [tex]\( y = -\frac{1}{7}x \)[/tex] shows a proportional relationship.
