Magan is deciding between two parking garages. Garage A charges an initial fee of $7 to park plus $3 per hour. Garage B charges an initial fee of $9 to park plus $2 per hour. Let AA represent the amount Garage A would charge if Magan parks for tt hours, and let BB represent the amount Garage B would charge if Magan parks for tt hours. Write an equation for each situation, in terms of t, commat, and determine the hours parked, t, commat, that would make the cost of each garage the same.

To solve this problem, we'll start by writing equations for the cost of parking at each garage in terms of $ t $, where $ t $ represents the number of hours Magan parks.

### Step 1: Write the equations for each garage.

1. **Garage A:**
   - The initial fee is $7.
   - The additional cost is $3 per hour.
   - The total cost, $ A(t) $, is given by:
   \[
   A(t) = 7 + 3t
   \]

2. **Garage B:**
   - The initial fee is $9.
   - The additional cost is $2 per hour.
   - The total cost, $ B(t) $, is given by:
   \[
   B(t) = 9 + 2t
   \]

### Step 2: Set the equations equal to each other to find the value of $ t $ that makes the costs equal.

To find when the costs are the same, set $ A(t) $ equal to $ B(t) $:

\[
7 + 3t = 9 + 2t
\]

### Step 3: Solve for $ t $.

1. Subtract $ 2t $ from both sides of the equation:
   \[
   7 + t = 9
   \]

2. Subtract 7 from both sides:
   \[
   t = 2
   \]

### Conclusion:

The cost of parking at both garages will be the same if Magan parks for **2 hours**.

Would you like more details or have any questions?

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Here are 5 relative questions to expand on this information:

1. How much would Magan pay at Garage A if she parks for 4 hours?
2. What is the total cost difference between the two garages if Magan parks for 3 hours?
3. How would the equations change if Garage A increased its hourly rate to $4 per hour?
4. At what parking duration would Garage B become more economical than Garage A?
5. If Magan parks for 5 hours, how much more expensive is Garage A compared to Garage B?

**Tip:** When comparing costs with different initial fees and rates, finding the breakeven point (where costs are equal) can help you decide the best option for different durations.

Magan is deciding between two parking garages. Garage A charges an initial fee of $7 to park plus $3 per hour. Garage B charges an initial fee of $9 to park plus $2 per hour. Let A represent the amount Garage A would charge if Magan parks for t hours, and let B represent the amount Garage B would charge if Magan parks for t hours. Write an equation for each situation, in terms of t, and determine the hours parked, t, that would make the cost of each garage the same.

Learn how to compare linear equations to determine when the cost of parking at two different garages will be the same. This example involves writing and solving linear equations based on initial fees and hourly rates, suitable for students in Grades 6-8.

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