The flat fee of a cab is $3.50 plus an additional $2.25 every mile.
b. Write an equation in slope-intercept, point-slope, or standard form. Explain why you chose the form you did
What does that mean?
Fixed Cost = $3.50
Variable Cost = $2.25 per mile
x = distance traveled in miles
Variable Cost = $2.25 ∙ x
y = Total Cost = Variable Cost + Fixed Cost
y = 2.25 x + 3.5
In this question, you are being asked to write an equation that represents the total cost of a cab ride, depending on the number of miles traveled. The three forms mentioned, slope-intercept form, point-slope form, and standard form, are different ways to express a linear equation.
1. Slope-intercept form: y = mx + b
2. Point-slope form: y - y1 = m(x - x1)
3. Standard form: Ax + By = C
The choice of which form to use depends on personal preference and the information available. In this case, since we know the flat fee and the additional cost per mile, it would be most convenient to use the slope-intercept form. This form allows us to directly interpret the intercept (flat fee) and the slope (additional cost per mile) of the equation.
To write an equation in slope-intercept, point-slope, or standard form, we need to understand what each of these forms represents.
1. Slope-intercept form: y = mx + b
In this form, 'm' represents the slope and 'b' represents the y-intercept of the equation.
2. Point-slope form: y - y₁ = m(x - x₁)
In this form, 'm' represents the slope, and (x₁, y₁) represents the coordinates of a point on the line.
3. Standard form: Ax + By = C
In this form, 'A,' 'B,' and 'C' are constants, and 'A' and 'B' cannot both be zero at the same time.
Now, let's write an equation for the given scenario:
The flat fee of a cab is $3.50, which means irrespective of the number of miles traveled, $3.50 is always added. Additionally, for every mile, an extra $2.25 is added.
Since the flat fee is constant and does not depend on the number of miles traveled, we can represent it as the y-intercept 'b'. So, b = 3.50.
For every mile traveled, an extra $2.25 is added. This indicates that the cost increases linearly with the number of miles. The rate of increase is the slope 'm', which is $2.25 per mile.
Therefore, the equation in slope-intercept form is: y = 2.25x + 3.50.
We choose slope-intercept form because it directly represents the slope and y-intercept given the problem's context. The slope represents the rate of increase, and the y-intercept represents the initial cost.