We are tasked with finding a linear equation $y = mx + b$ to represent the total monthly cost, where $x$ is the number of minutes used, and $y$ is the total monthly cost.

Part A: Finding the Equation $y = mx + b$

We are given two points that describe the relationship between minutes used and monthly cost:

• When $x = 290$ minutes, $y = 169$ dollars.
• When $x = 770$ minutes, $y = 409$ dollars.

These two points are: $(290, 169)$ and $(770, 409)$.

Step 1: Find the slope $m$

The slope of the line is calculated using the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substituting the given values: $m = \frac{409 - 169}{770 - 290} = \frac{240}{480} = 0.5$ Thus, the slope $m = 0.5$.

Step 2: Find the y-intercept $b$

Now that we know the slope, we can use one of the points to find the y-intercept. Using the point $(290, 169)$ and the equation $y = mx + b$: $169 = 0.5(290) + b$ $169 = 145 + b$ $b = 169 - 145 = 24$ Thus, the y-intercept $b = 24$.

Final Equation:

The equation that models the total monthly cost is: $y = 0.5x + 24$

Part B: Find the total monthly cost if 754 minutes are used

Now, we use the equation $y = 0.5x + 24$ to find the cost when $x = 754$.

Substitute $x = 754$ into the equation: $y = 0.5(754) + 24$ $y = 377 + 24 = 401$

Thus, if 754 minutes are used, the total monthly cost will be $401.

Summary:

• Equation: $y = 0.5x + 24$
• Total Cost for 754 minutes: $401

Would you like more details on any part of this explanation?

Here are 5 related questions to explore this concept further:

1. How would the equation change if the cost per minute increased?
2. What happens to the total monthly cost if no minutes are used?
3. How can we graph this equation and interpret its meaning?
4. If a customer uses 1000 minutes, what would their total cost be?
5. How would the equation change if the flat monthly fee increased?

Tip: In any linear equation, the slope $m$ tells you how much the dependent variable (y) changes for each unit increase in the independent variable (x).