A phone company has a monthly cellular data plan where a...

A phone company has a monthly cellular data plan where a customer pays a flat monthly fee of $\$ 10$ and then a certain amount of money per megabyte $(\mathrm{MB})$ of data used on the phone. If a customer uses 20 $\mathrm{MB}$ , the monthly cost will be $\$ 11.20$ . If the customer uses 130 $\mathrm{MB}$ , the monthly cost will be $\$ 17.80$ . a. Find a linear equation for the monthly cost of the data plan as a function of $x$ , the number of MB used. b. Interpret the slope and $y$ -intercept of the equation. c. Use your equation to find the total monthly cost if 250 $\mathrm{MB}$ are used.

A phone company has a monthly cellular data plan where a customer pays a flat monthly fee of $\$ 10$ and then a certain amount of money per megabyte $(\mathrm{MB})$ of data used on the phone. If a customer uses 20 $\mathrm{MB}$ , the monthly cost will be $\$ 11.20$ . If the customer uses 130 $\mathrm{MB}$ , the monthly cost will be $\$ 17.80$ .

a. Find a linear equation for the monthly cost of the data plan as a function of $x$ , the number of MB used.
b. Interpret the slope and $y$ -intercept of the equation.
c. Use your equation to find the total monthly cost if 250 $\mathrm{MB}$ are used.

Key Concepts

Linear Equations

Linear equations express a constant rate of change between two quantities. They form a straight-line graph when plotted on a coordinate plane and are typically written in the form y = mx + b, where m represents the slope and b the y-intercept.

Constructing a Linear Equation from Two Data Points

Given two pairs of corresponding values, one can determine the linear relationship by calculating the slope (rate of change) and then using one of the points to solve for the y-intercept. This method is essential in creating a model that accurately represents the relationship between the variables.

Slope-Intercept Form

The slope-intercept form, y = mx + b, is a commonly used representation of a linear function. In this format, 'm' denotes the rate at which the dependent variable changes with respect to the independent variable, while 'b' indicates the value of the dependent variable when the independent variable is zero.

Interpreting the Slope

The slope in a linear equation quantifies the change in the dependent variable for each unit increase in the independent variable. It represents the constant rate at which one variable responds to changes in the other, making it key for understanding the relationship between the two quantities.

Interpreting the Y-Intercept

The y-intercept is the point where the linear graph crosses the y-axis, meaning it shows the initial value of the dependent variable when the independent variable is zero. This value often represents a fixed or starting cost before any variable costs are added.

Using Linear Models for Predictions

Once a linear equation is formulated, it can be used to predict outcomes for any given value of the independent variable within the model's range. This predictive capability is crucial for estimating results and making informed decisions based on the established relationship.

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