Choc-Arcade has Lucy's two favorite things: a chocolate shop and an arcade! Not only that, but Choc-Arcade's loyalty program rewards Lucy with free game tokens every time she buys chocolates.

There is a proportional relationship between the number of chocolates Lucy buys, [tex]x[/tex], and the number of free game tokens she gets, [tex]y[/tex].

[tex]
\[
\begin{array}{|c|c|}
\hline
x \, (\text{chocolates}) & y \, (\text{tokens}) \\
\hline
3 & 12 \\
\hline
13 & 52 \\
\hline
16 & 64 \\
\hline
20 & 80 \\
\hline
\end{array}
\]
[/tex]

Write an equation for the relationship between [tex]x[/tex] and [tex]y[/tex]. Simplify any fractions.

[tex] y = \square [/tex]

Since the relationship is proportional, we can express it with the equation

[tex]$$
y = kx,
$$[/tex]

where [tex]$k$[/tex] is the constant of proportionality.

1. Choose one of the data points to find [tex]$k$[/tex]. For example, using the point [tex]$(3, 12)$[/tex], we have

[tex]$$
k = \frac{12}{3} = 4.
$$[/tex]

2. Now, verify that this same value of [tex]$k$[/tex] works for the other points:
- For the point [tex]$(13, 52)$[/tex]:

[tex]$$
k = \frac{52}{13} = 4.
$$[/tex]

- For the point [tex]$(16, 64)$[/tex]:

[tex]$$
k = \frac{64}{16} = 4.
$$[/tex]

- For the point [tex]$(20, 80)$[/tex]:

[tex]$$
k = \frac{80}{20} = 4.
$$[/tex]

Since all points give the same constant [tex]$k = 4$[/tex], the equation that relates [tex]$x$[/tex] and [tex]$y$[/tex] is

[tex]$$
y = 4x.
$$[/tex]