Cab companies often charge a flat fee for picking someone up and then charge an additional fee per mile driven
The city of Charlotte, NC charges $2.50 fee and $2.50 per Mile for each cab ride
Write an equation in point-slope form to represent the amount charged, C, for m number of miles by the company in Charlotte. (Hint: Go to Desmos and convert the equation you created in #1 to a table. Use a point from the table)
In point-slope form, the equation of a line can be written as:
\[ y - y_1 = m(x - x_1) \]
where \( m \) is the slope of the line, and \( (x_1, y_1) \) is a point on the line.
For the cab fare situation in Charlotte, NC, the amount charged, \( C \), can be considered as the dependent variable \( y \), while the number of miles, \( m \), can be considered as the independent variable \( x \).
We know the following:
- There is a flat fee of $2.50, which is the y-intercept (\( b \)) of our linear equation.
- The charge per mile is $2.50, which is our slope (\( m \)).
If we didn't start from zero miles, we could use the point-slope form with an arbitrary point, however, since we have no other point given, we could use the y-intercept as our point, since a taxi ride starts from no mileage. Thus, after the flat fee, our point can be \( (0, 2.50) \).
The slope, \( m \), is the additional fee per mile, which is $2.50 per mile. So we can plug the slope and the point into the point-slope formula:
\[ C - 2.50 = 2.50(m - 0) \]
Simplified, the equation becomes:
\[ C - 2.50 = 2.50m \]
This equation represents the amount charged \( C \) for \( m \) number of miles by the cab company in Charlotte. To write it in a more simplified way, we can add $2.50 to both sides to solve for \( C \):
\[ C = 2.50m + 2.50 \]
Although this final form isn't in point-slope form, it is the same linear relationship that shows the cab fare based on miles driven, starting with a flat fee. If you stick to the point-slope form as initially asked, it would just be the equation \( C - 2.50 = 2.50m \).