a mobile phone company charges a fixed monthly fee of $20 plus $0.10 per minute of calls.
a. write a linear equation to represent the total monthly cost y based on the number of minutes x used
b. graph the equation and use the graph to estimate the monthly cost for 150 minutes of calls
c. explain how the slope and y-intercept relate to the charges in this context

Let's break down the problem step by step.

### a. Writing the Linear Equation

The total monthly cost $ y $ depends on two factors:
1. A fixed monthly fee of $20.
2. A variable cost of $0.10 per minute of calls, where $ x $ is the number of minutes used.

The linear equation that represents this relationship is:

\[
y = 0.10x + 20
\]

### b. Graphing the Equation

To graph the equation $ y = 0.10x + 20 $:

1. **Plot the y-intercept**: Start at the point $ (0, 20) $ because when $ x = 0 $, $ y = 20 $.
2. **Use the slope**: The slope is 0.10, which means for every additional minute, the cost increases by $0.10.

To estimate the monthly cost for 150 minutes of calls, substitute $ x = 150 $ into the equation:

\[
y = 0.10(150) + 20 = 15 + 20 = 35
\]

So, the estimated monthly cost for 150 minutes of calls is $35.

### c. Explanation of the Slope and Y-Intercept

- **Slope (0.10)**: This represents the cost per minute of calls. For every additional minute, the total cost increases by $0.10.
  
- **Y-intercept (20)**: This represents the fixed monthly fee. Even if no calls are made (i.e., $ x = 0 $), the cost will still be $20.

The slope tells you how much the cost changes with each additional minute, while the y-intercept gives the base cost before any calls are made.

Would you like to see the graph of the equation, or do you have any questions? 

Here are 5 relative questions for you:

1. How would the equation change if the per-minute cost was $0.15 instead of $0.10?
2. What would the y-intercept be if there were no fixed monthly fee?
3. How would the graph change if the fixed monthly fee increased to $30?
4. What does it mean if the slope of the line was steeper?
5. How can you use this equation to predict the cost for any number of minutes?

**Tip:** When interpreting a linear equation, the slope always represents the rate of change, while the y-intercept represents the initial value before any changes occur.

Understand how to calculate the monthly cost of phone calls using a linear equation. Learn about the fixed monthly fee and the per-minute charge, graph the equation, and estimate costs for specific call durations. Ideal for students in Grades 7-9 studying linear equations and applications.

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