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We are tasked with determining the one-time registration charge and the monthly fee based on the following information:

- After 4 months, Tom paid $420.
- After 9 months, Tom paid $795.

### Step-by-step approach:

Let:
- $ R $ be the one-time registration charge.
- $ M $ be the monthly fee.

Using the information given:

1. After 4 months: 
   \[
   R + 4M = 420
   \]

2. After 9 months:
   \[
   R + 9M = 795
   \]

### Solving the system of equations:

We have two equations:
1. $ R + 4M = 420 $
2. $ R + 9M = 795 $

Let's subtract the first equation from the second:
\[
(R + 9M) - (R + 4M) = 795 - 420
\]
This simplifies to:
\[
5M = 375
\]
Solving for $ M $:
\[
M = \frac{375}{5} = 75
\]

Now, substitute $ M = 75 $ into the first equation:
\[
R + 4(75) = 420
\]
\[
R + 300 = 420
\]
\[
R = 420 - 300 = 120
\]

### Final answer:
- The registration charge $ R $ is **$120**.
- The monthly fee $ M $ is **$75**.

Would you like more details or have any questions?

Here are some related questions:
1. What would Tom have paid after 12 months?
2. How does increasing the monthly fee affect the total payment over time?
3. If the monthly fee was reduced by $10, what would be the new registration fee?
4. What happens to the registration fee if the monthly fee was doubled?
5. How can you solve such systems of equations graphically?

**Tip:** When solving systems of equations, isolating one variable first often makes the problem more manageable.

Tom pays a one-time registration charge and regular monthly fees to belong to a fitness club. After four months, he had paid $420. After nine months, he had paid $795. Determine the registration charge and the monthly fee.

Learn how to determine the one-time registration fee and monthly charges for a fitness club membership based on total payments after 4 and 9 months. This example covers solving systems of linear equations, ideal for students in grades 8-10.

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