Questions: The total cost for a bucket of popcorn and 4 movie tickets is 56. The total cost for the same size bucket of popcorn and 6 movie tickets is 80. The cost of a bucket of popcorn is 8. Which equation represents the relationship between y, the total cost of the popcorn and movie tickets, and x, the number of movie tickets that are purchased? y=12x+8 y=12x-8 y=14x+8 y=14x-8

The total cost for a bucket of popcorn and 4 movie tickets is 56. The total cost for the same size bucket of popcorn and 6 movie tickets is 80. The cost of a bucket of popcorn is 8. Which equation represents the relationship between y, the total cost of the popcorn and movie tickets, and x, the number of movie tickets that are purchased?
y=12x+8
y=12x-8
y=14x+8
y=14x-8

Transcript text: The total cost for a bucket of popcorn and 4 movie tickets is $\$ 56$. The total cost for the same size bucket of popcorn and 6 movie tickets is $\$ 80$. The cost of a bucket of popcorn is $\$ 8$. Which equation represents the relationship between $y$, the total cost of the popcorn and movie tickets, and $x$, the number of movie tickets that are purchased? $y=12 x+8$ $y=12 x-8$ $y=14 x+8$ $y=14 x-8$

Solution

Solution Steps

Step 1: Define Variables

Let:

$ y $ = total cost of popcorn and movie tickets
$ x $ = number of movie tickets purchased
$ p $ = cost of a bucket of popcorn = \$8
$ t $ = cost of one movie ticket

Step 2: Set Up Equations

From the problem, we have two scenarios:

For 4 movie tickets and a bucket of popcorn: \[ y = p + 4t = 56 \] Substituting $ p = 8 $: \[ 8 + 4t = 56 \]
For 6 movie tickets and a bucket of popcorn: \[ y = p + 6t = 80 \] Substituting $ p = 8 $: \[ 8 + 6t = 80 \]

Step 3: Solve for $ t $

From the first equation: \[ 8 + 4t = 56 \\ 4t = 56 - 8 \\ 4t = 48 \\ t = 12 \]

From the second equation: \[ 8 + 6t = 80 \\ 6t = 80 - 8 \\ 6t = 72 \\ t = 12 \]

Both equations give $ t = 12 $, so the cost of one movie ticket is \$12.

Step 4: Formulate the Relationship

The total cost $ y $ is the sum of the cost of the popcorn and the cost of the movie tickets: \[ y = p + t \cdot x \] Substituting $ p = 8 $ and $ t = 12 $: \[ y = 8 + 12x \]