Harry's soccer team plays 2 nonconference games for every 3 conference games.

If [tex]\( y \)[/tex] represents the number of nonconference games and [tex]\( x \)[/tex] represents the number of conference games, which equation best models this proportional relationship?

A. [tex]\( y = \frac{2}{3} x \)[/tex]
B. [tex]\( y = \frac{3}{2} x \)[/tex]
C. [tex]\( y = 2x \)[/tex]
D. [tex]\( y = 6x \)[/tex]



Answer :

To determine the equation that best models the proportional relationship between the number of nonconference games, [tex]\( y \)[/tex], and the number of conference games, [tex]\( x \)[/tex], we start by examining the ratio given in the problem.

Harry’s soccer team plays 2 nonconference games for every 3 conference games. This can be written as the ratio:

[tex]\[ \frac{y}{x} = \frac{2}{3} \][/tex]

To express [tex]\( y \)[/tex] (nonconference games) as a function of [tex]\( x \)[/tex] (conference games), we can use the concept of direct proportion. Proportional relationships can be represented by equations of the form:

[tex]\[ y = kx \][/tex]

where [tex]\( k \)[/tex] is the constant of proportionality. In this problem, [tex]\( k \)[/tex] is given by the ratio of the number of nonconference games to the number of conference games:

[tex]\[ k = \frac{2}{3} \][/tex]

Thus, the equation that best models this relationship is:

[tex]\[ y = \frac{2}{3} x \][/tex]

To confirm, let's summarize:
- The ratio [tex]\(\frac{y}{x} = \frac{2}{3}\)[/tex] indicates that for every 3 conference games, there are 2 nonconference games.
- Representing this proportional relationship as [tex]\( y \)[/tex], which stands for the number of nonconference games, in terms of [tex]\( x \)[/tex], which stands for the number of conference games, we find the constant of proportionality to be [tex]\(\frac{2}{3}\)[/tex].

Therefore, the correct equation is:

[tex]\[ y = \frac{2}{3} x \][/tex]

This is confirmed by the proportional relationship given in the problem.