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To determine when the total cost for both health clubs will be the same, we can set up a system of linear equations based on the membership fees and monthly fees provided.

For the first club:
- Membership cost: $22.00
- Monthly fee: $25.75

So, the total cost for the first club after $ x $ months is:
\[
C_1(x) = 22 + 25.75x
\]

For the second club:
- Membership cost: $47.00
- Monthly fee: $19.50

The total cost for the second club after $ x $ months is:
\[
C_2(x) = 47 + 19.50x
\]

Now, set the two cost equations equal to each other to find when the costs are the same:
\[
22 + 25.75x = 47 + 19.50x
\]

Now, solve for $ x $:

1. Subtract $ 19.50x $ from both sides:
\[
22 + 6.25x = 47
\]

2. Subtract 22 from both sides:
\[
6.25x = 25
\]

3. Divide by 6.25:
\[
x = 4
\]

So, after **4 months**, the total cost for both health clubs will be the same.

Would you like a further breakdown or details on this?

Here are 5 related questions you can explore:
1. How would the comparison change if one of the clubs offered a discount after a certain number of months?
2. What is the total cost for each club after 6 months?
3. How can you represent this problem graphically?
4. How would the solution change if only the monthly fee increased for both clubs?
5. What if Veronica could cancel her membership after 3 months? How would the costs compare?

**Tip:** When comparing costs over time, focus on both fixed initial fees and recurring fees to see when they balance out.

Veronica is choosing between two health clubs. After how many months will the total cost for each health club be the same?

Solve the problem of finding after how many months the total cost of two health clubs will be the same. This involves setting up and solving linear equations based on membership fees and monthly rates. Suitable for students in Grades 6-8.

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