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.

(5 points)

0 / 10000 Word Limit
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Answer 1

The customer's approach has a mistake in the equations used to represent the costs of renting cars from Zippy Rent-a-Car and Speedy Rent-a-Car. Let's identify and correct that to find the point at which the costs are the same.

Definitions and Correct Equations

Let:

• \( x \) = number of days the car is rented.
• \( C \) = total cost for each rental company.

For Zippy Rent-a-Car: The total cost formula is: \[ C = 30 + 50x \] where \( 30 \) is the one-time fee and \( 50x \) is the daily rental cost for \( x \) days.

For Speedy Rent-a-Car: The total cost formula is: \[ C = 70 + 40x \] where \( 70 \) is the one-time fee and \( 40x \) is the daily rental cost for \( x \) days.

Step-by-Step Solution

Step 1: Set the equations equal to each other to find when the costs are the same:

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

Step 2: Subtract \( 40x \) from both sides:

\[ 30 + 10x = 70 \]

Step 3: Subtract \( 30 \) from both sides:

\[ 10x = 40 \]

Step 4: Divide by \( 10 \):

\[ x = 4 \]

Conclusion

After correcting the equations, we find that the costs for both rental companies will be the same after 4 days. The customer's initial conclusion of \( \frac{1}{4} \) day was incorrect due to an error in writing the cost equations. The rental comparisons must factor in the one-time fees and the appropriate coefficients for the rental prices as established above.