The total cost for a bucket of popcom and 4 movie tickets is \( \$ 56 \). The total cost for the same size bucket of popcom and 6 movic tickets is \( \$ 80 \). The cost of a bucket of popcorn is \( \$ 8 \). Which equation represcnts the relationship benween \( y \) the total cost of the popcorn and movie tickets, and \( x \), the number of movie tickets that are purchased? \( y=12 x+8 \) \( y=12 x-8 \) \( y=14 x+8 \) \( y=14 x-8 \)

To determine the relationship between \( y \) (the total cost) and \( x \) (the number of movie tickets), let's break down the problem step by step. 1. **Given Information:** - Cost of a bucket of popcorn (\( p \)): \$8 - Total cost for popcorn and 4 movie tickets: \$56 - Total cost for popcorn and 6 movie tickets: \$80 2. **Let \( t \) be the cost of one movie ticket.** 3. **Set Up the Equations:** - From the first scenario: \[ p + 4t = 56 \] Substituting \( p = 8 \): \[ 8 + 4t = 56 \implies 4t = 48 \implies t = 12 \] - From the second scenario (to verify): \[ p + 6t = 80 \] Substituting \( p = 8 \) and \( t = 12 \): \[ 8 + 6(12) = 8 + 72 = 80 \quad \text{(which matches)} \] 4. **Formulate the Equation:** - The total cost \( y \) is the sum of the cost of popcorn and the cost of movie tickets: \[ y = p + xt \] Substituting \( p = 8 \) and \( t = 12 \): \[ y = 8 + 12x \] Which can also be written as: \[ y = 12x + 8 \] **Answer:** \( y = 12x + 8 \)