Question

select the equations that show a proportional relationship between x and y.
☑️ ( y = 9x )
☑️ ( y = 3x + 9 )

Explanation:

To determine which equations show a proportional relationship between \( x \) and \( y \), we recall that a proportional relationship has the form \( y = kx \), where \( k \) is a constant (the constant of proportionality) and the line passes through the origin \((0,0)\) (i.e., when \( x = 0 \), \( y = 0 \)).

Analyzing \( y = 9x \):

For \( y = 9x \), if we substitute \( x = 0 \), we get \( y = 9(0) = 0 \). This means the line passes through the origin, and it is in the form \( y = kx \) (with \( k = 9 \)). Thus, \( y = 9x \) represents a proportional relationship.

Analyzing \( y = 3x + 9 \):

For \( y = 3x + 9 \), if we substitute \( x = 0 \), we get \( y = 3(0) + 9 = 9 \). This means the line does not pass through the origin (it has a \( y \)-intercept of \( 9 \)) and is in the form \( y = mx + b \) (where \( b
eq 0 \)). A proportional relationship requires \( b = 0 \), so \( y = 3x + 9 \) does not represent a proportional relationship.

Correcting the Selection:

The only equation with a proportional relationship is \( y = 9x \). The equation \( y = 3x + 9 \) should not be selected.

Answer:

The correct equation showing a proportional relationship is \( \boldsymbol{y = 9x} \). The equation \( y = 3x + 9 \) does not represent a proportional relationship (so it should be deselected).