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.
