Aria has a points card for a movie theater. - She receives 70 rewards points just for signing up. - She earns \( 4-5 \) points for each visit to the movie theater. - She needs 133 points for a free movie ticket. Which equation could be used to determine \( x \), the number of visits Aria must make to earn a free movie ticket? Answer Attempt 1 out of 3 \( x=\frac{70-133}{4.5} \) \( 70 x+4.5=133 \)

To determine the number of visits \( x \) Aria needs to make to earn a free movie ticket, let's break down the information provided: 1. **Initial Points:** Aria starts with **70 points** just for signing up. 2. **Points per Visit:** She earns between **4 to 5 points** for each visit. For calculation purposes, we'll use the average, **4.5 points per visit**. 3. **Total Points Needed:** She needs **133 points** for a free movie ticket. ### Setting Up the Equation The total points Aria accumulates can be expressed as the sum of her initial points and the points earned from visiting the theater: \[ \text{Total Points} = \text{Initial Points} + (\text{Points per Visit} \times \text{Number of Visits}) \] Plugging in the values: \[ 133 = 70 + 4.5x \] ### Solving for \( x \) \[ 4.5x = 133 - 70 \] \[ 4.5x = 63 \] \[ x = \frac{63}{4.5} \] \[ x = 14 \] ### Conclusion Aria needs to make **14 visits** to the movie theater to earn a free movie ticket. ### Correct Equation The correct equation to determine \( x \) is: \[ 70 + 4.5x = 133 \] --- **Feedback on Your Attempts:** 1. **\( x = \frac{70 - 133}{4.5} \):** - This equation incorrectly subtracts 133 from 70, resulting in a negative number, which doesn't make sense in this context. 2. **\( 70x + 4.5 = 133 \):** - This equation incorrectly multiplies the initial points (70) by the number of visits \( x \), which isn't accurate. The initial 70 points should be added to the points earned from visits, not multiplied. The correct approach is to **add** the initial points to the points earned from visits, as shown in the equation above.