- Identify the linear relationship as \(y = mx + b\) and use the given points \((2, 4.25)\) and \((7, 8.50)\).
- Calculate the slope as \(m = \frac{8.50 - 4.25}{7 - 2} = 0.85\).
- Find the y-intercept \(b\) by substituting a point into the equation: \(4.25 = 0.85(2) + b\), giving \(b = 2.55\).
- State the final linear model: \(\boxed{y = 0.85x + 2.55}\).
Explanation
- Problem Analysis
We are given that a cab ride cost \(y\) is a function of the distance \(x\) in miles, and this relationship is linear. We are provided with two data points: a 2-mile ride costs \(\$4.25\), represented by the point \((2, 4.25)\), and a 7-mile ride costs \(\$8.50\), represented by the point \((7, 8.50)\). We need to determine the linear equation in the form \(y = mx + b\) that fits these two points.
- Calculating the Slope
The slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
Substituting the given points \((2, 4.25)\) and \((7, 8.50)\):
\[m = \frac{8.50 - 4.25}{7 - 2} = \frac{4.25}{5} = 0.85\]
So, the slope of the line is \(0.85\).
- Finding the Y-Intercept
Now that we have the slope \(m = 0.85\), we can use the equation \(y = 0.85x + b\) and one of our data points, for example \((2, 4.25)\), to find the y-intercept \(b\):
\[4.25 = 0.85(2) + b\]
\[4.25 = 1.70 + b\]
Subtracting 1.70 from both sides:
\[b = 4.25 - 1.70 = 2.55\]
Thus, the y-intercept is \(2.55\).
- Constructing the Final Equation
With the slope \(m = 0.85\) and the y-intercept \(b = 2.55\), the linear equation modeling the cost of the cab ride is:
\[y = 0.85x + 2.55\]
This matches the third option provided. The final answer is \(\boxed{y = 0.85x + 2.55}\).
Examples
Linear modeling is used in various real-life scenarios, such as calculating service fees. For example, a home technician might charge a fixed call-out fee and an hourly labor rate. By representing this as \(y = mx + b\), where \(b\) is the call-out fee, \(m\) is the hourly rate, and \(x\) is the number of hours worked, you can easily calculate the total cost for any length of service. Understanding these equations helps in budgeting and evaluating service costs.