To find the equation that represents the relationship between [tex]\( y \)[/tex] (the total cost of the popcorn and movie tickets) and [tex]\( x \)[/tex] (the number of movie tickets purchased), we need to break down the information given and follow a step-by-step approach:
1. Identify the known variables:
- Total cost for popcorn and 4 movie tickets is \[tex]$56.
- Total cost for popcorn and 6 movie tickets is \$[/tex]80.
- The cost of a bucket of popcorn is \[tex]$8.
2. Set up the relationship in terms of movie tickets:
Let's denote the cost of one movie ticket as \( C \).
3. Establish two equations based on the given information:
- For 4 movie tickets and 1 bucket of popcorn:
\[
4C + 8 = 56
\]
- For 6 movie tickets and 1 bucket of popcorn:
\[
6C + 8 = 80
\]
4. Solve the first equation for \( C \):
\[
4C + 8 = 56
\]
Subtract 8 from both sides:
\[
4C = 48
\]
Divide by 4:
\[
C = 12
\]
5. Verify \( C \) with the second equation:
\[
6C + 8 = 80
\]
Substitute \( C = 12 \) into the equation:
\[
6(12) + 8 = 80
\]
\[
72 + 8 = 80
\]
The verification confirms \( C = 12 \) or \$[/tex]12 per movie ticket.
6. Formulate the equation [tex]\( y \)[/tex]:
- [tex]\( y \)[/tex] is the total cost of [tex]\( x \)[/tex] number of movie tickets and 1 bucket of popcorn.
[tex]\[
y = 12x + 8
\][/tex]
Thus, the equation representing the relationship between [tex]\( y \)[/tex] and [tex]\( x \)[/tex] is:
[tex]\[
y = 12x + 8
\][/tex]
Therefore, the correct option is [tex]\( y = 12x + 8 \)[/tex].
