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 $$ x=\frac{70-133}{4.5} $$ $$ 70 x+4.5=133 $$

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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.
To find the number of visits $$ x $$ Aria needs to make to earn 133 points, use the equation: $$ 70 + 4.5x = 133 $$ Solving this gives $$ x = 14 $$ visits.

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