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Marques has a points card for a movie theater.

He receives 70 rewards points just for signing up.
He earns 9.5 points for each visit to the movie theater.
He needs 184 points for a free movie ticket.

Write and solve an equation which can be used to determine v, the number of visits Marques must make to earn a free movie ticket.

3 answers

To solve for the number of visits \( v \) that Marques must make to earn a free movie ticket, we can set up an equation based on the points he receives.

  1. Marques receives 70 points for signing up.
  2. He earns 9.5 points for each visit.
  3. He needs a total of 184 points for a free movie ticket.

The total points Marques will have after \( v \) visits can be expressed as: \[ \text{Total Points} = 70 + 9.5v \]

We can set this expression equal to the points needed for a free movie ticket: \[ 70 + 9.5v = 184 \]

Now, we can solve for \( v \):

  1. Subtract 70 from both sides: \[ 9.5v = 184 - 70 \] \[ 9.5v = 114 \]

  2. Divide both sides by 9.5 to isolate \( v \): \[ v = \frac{114}{9.5} \]

  3. Now perform the division: \[ v \approx 12 \]

So, the number of visits \( v \) that Marques must make to earn a free movie ticket is approximately \( 12 \).

To verify: If he makes 12 visits, he earns: \[ 9.5 \times 12 = 114 \text{ points from visits.} \] Adding the initial 70 points he has: \[ 70 + 114 = 184 \text{ points.} \] This confirms that he needs 12 visits to reach the required 184 points.

what about in Equation form

Certainly! We can express the solution in equation form as follows.

  1. We start with the equation that represents the total points Marques has after \( v \) visits to the movie theater: \[ 70 + 9.5v = 184 \]

  2. The above equation can also be rearranged for clarity: \[ 9.5v = 184 - 70 \]

  3. Simplifying that gives: \[ 9.5v = 114 \]

  4. Finally, we can express \( v \) in terms of the other values: \[ v = \frac{114}{9.5} \]

This maintains the original equation form, leading to the solution: \[ v = 12 \]

Thus, the final equation used to find \( v \) is: \[ 70 + 9.5v = 184 \]