At Zippy Rent-a-Car, you can rent a car for $50 per day with a one-time fee of $30. At Speedy Rent-a-Car, you can rent a car for $40 per day with a one-time fee of $70. The goal is to find the number of days for which both rental locations will cost the same amount using substitution.

Question

System of Equations:

Let x represent the number of days the car is rented, and let C represent the total cost for each rental company.

Zippy Rent-a-Car: C= 30x+50
Speedy Rent-a-Car: C= 70x+40
The customer used substitution to find when the costs would be the same. Here is their solution process:

Step 1: Set the equations equal to each other (since the total costs should be the same). 30x+50=70x+40

Step 2: Subtract 30x from both sides. 50 = 40x + 40

Step 3: Subtract 40 from both sides. 10=40x

Step 4: Divide by 40. x=1/4

The customer concluded that the rental costs would be the same after 1/4 day.

Task:
Identify and explain the mistake in the customer's approach, if any. Then, correctly solve the system of equations using substitution to determine the number of days at which both rental companies will charge the same total amount.

Answer 1

The customer's approach contains a mistake in the formulation of the cost equations. The equations for the total cost for each rental company should be structured as follows:

For Zippy Rent-a-Car: \[ C = 30 + 50x \] (The $30 is the one-time fee and the $50 is the daily rental rate, multiplied by the number of days \(x\).)

For Speedy Rent-a-Car: \[ C = 70 + 40x \] (Again, here the $70 is the one-time fee and the $40 is the daily rental rate, multiplied by the number of days \(x\).)

Now, let's set the two equations equal to each other to find out when both costs are the same:

\[ 30 + 50x = 70 + 40x \]

Next, we'll solve for \(x\):

Step 1: Subtract \(40x\) from both sides:

\[ 30 + 10x = 70 \]

Step 2: Subtract 30 from both sides:

\[ 10x = 40 \]

Step 3: Divide by 10:

\[ x = 4 \]

So the correct solution indicates that the rental costs will be the same after 4 days, not 1/4 day.

Summary of the mistake:

1. The customer incorrectly set up the equations for total cost by swapping the one-time fee and the daily rental rate.
2. The substitution approach was not employed correctly with the proper cost equations, leading to an incorrect conclusion of 1/4 day instead of the correct answer of 4 days.