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)

Answer 1

The mistake in the customer's approach lies in the formulation of the cost equations. The customer incorrectly formulated Zippy Rent-a-Car's equation. The correct equations for the total cost (C) for each rental car company should be:

• For Zippy Rent-a-Car: \( C = 30 + 50x \) (the one-time fee of $30 plus $50 for each day rented)
• For Speedy Rent-a-Car: \( C = 70 + 40x \) (the one-time fee of $70 plus $40 for each day rented)

Thus, the correct equations are:

1. \( C = 30 + 50x \) (Zippy Rent-a-Car)
2. \( C = 70 + 40x \) (Speedy Rent-a-Car)

Now, let's solve these equations correctly for the number of days \( x \) at which both rental companies would charge the same total amount.

Step 1: Set the equations equal to each other.

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

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

\[ 30 + 10x = 70 \]

Step 3: Subtract 30 from both sides.

\[ 10x = 40 \]

Step 4: Divide both sides by 10.

\[ x = 4 \]

Conclusion

Both rental companies will charge the same total amount after 4 days.