To determine the number of visits \( v \) that Ariana must make to earn a free movie ticket, we can set up the following equation based on the points she accumulates:
- She starts with 25 points (from signing up).
- For each visit, she earns 8.5 points.
- She needs a total of 76 points for a free movie ticket.
We can express the total points she earns after \( v \) visits as:
\[ \text{Total points} = 25 + 8.5v \]
We want this to equal 76 points (the points needed for a free ticket):
\[ 25 + 8.5v = 76 \]
Now, we can solve for \( v \).
- Subtract 25 from both sides:
\[ 8.5v = 76 - 25 \] \[ 8.5v = 51 \]
- Now, divide both sides by 8.5:
\[ v = \frac{51}{8.5} \]
- Calculating \( \frac{51}{8.5} \):
\[ v = 6 \]
Thus, the equation is \( 25 + 8.5v = 76 \) and the solution for \( v \) is:
\[ \boxed{6} \]
Ariana must make 6 visits to earn a free movie ticket.