An overseas phone call is charged at one rate (a fixed amount) for the first minute, and charged at a different rate for each additional minute. If a 7 minute phone call costs $10, and a 4 minute phone call costs $6.40, what is the rate charged for the first minute, and what is the rate charged for each minute after the first?

To solve this problem, we will set up a system of linear equations based on the information given about the phone call charges.

Let's define:
- $ x $ as the rate charged for the first minute.
- $ y $ as the rate charged for each additional minute after the first.

### Step-by-step solution:

1. **Set up the equations based on the given information:**

   - For a 7-minute phone call that costs $10:
     \[
     x + 6y = 10
     \]
     (since the first minute costs $x$ and the remaining 6 minutes cost $6y$).

   - For a 4-minute phone call that costs $6.40:
     \[
     x + 3y = 6.40
     \]
     (since the first minute costs $x$ and the remaining 3 minutes cost $3y$).

2. **Write down the system of equations:**

   \[
   \begin{cases}
   x + 6y = 10 \\
   x + 3y = 6.40
   \end{cases}
   \]

3. **Subtract the second equation from the first to eliminate $x$:**

   \[
   (x + 6y) - (x + 3y) = 10 - 6.40
   \]

   Simplifying this equation gives:

   \[
   3y = 3.60
   \]

4. **Solve for $y$:**

   \[
   y = \frac{3.60}{3} = 1.20
   \]

   So, the rate charged for each additional minute after the first is **$1.20**.

5. **Substitute $y = 1.20$ back into one of the original equations to solve for $x$:**

   Using the second equation:

   \[
   x + 3(1.20) = 6.40
   \]

   \[
   x + 3.60 = 6.40
   \]

   \[
   x = 6.40 - 3.60 = 2.80
   \]

   So, the rate charged for the first minute is **$2.80**.

### Conclusion:

- **Rate for the first minute** ($x$): **$2.80**
- **Rate for each additional minute** ($y$): **$1.20**

Would you like further details or have any questions?

Here are 5 related questions for further exploration:

1. What would be the cost of a 10-minute phone call at these rates?
2. How would the equations change if the rates for the first and additional minutes were reversed?
3. If a different phone plan charged $3.00 for the first minute and $0.80 for each additional minute, how would the cost of a 7-minute call compare?
4. How do these rates affect the cost of longer phone calls, say 15 or 20 minutes?
5. Can you derive a general formula for the cost of a phone call based on any given $x$ and $y$?

**Tip:** When solving systems of equations, check your solutions by substituting the values back into the original equations to ensure accuracy.

Learn how to calculate phone call rates for the first minute and each additional minute based on given examples. Solve the system of linear equations to find the rates. Includes examples of 7-minute and 4-minute calls costing $10 and $6.40, respectively. Suitable for Grades 7-9.

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