Question

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.

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 contains a mistake in setting up and solving the equations for the rental costs. Let's break down where the error occurred and then solve it correctly.

### Mistake in the Customer's Approach

In the initial equations provided, the cost for Zippy Rent-a-Car is given as \( C = 30 + 50x \) (30 is the one-time fee and 50 is the daily cost multiplied by the number of days \( x \)), while the cost for Speedy Rent-a-Car is \( C = 70 + 40x \).

The customer incorrectly set up the equation as:

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

This was an error in how the costs were represented. The correct cost equations should have been:

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

### Correctly Solve the System of Equations

Now let's set the correct equations equal to each other:

1. \( 30 + 50x = 70 + 40x \)

Now, 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:

\[ x = 4 \]

### Conclusion

The correct value of \( x \) indicates that both rental companies will charge the same total amount after **4 days**.

Thus, the mistake was in the initial representation of the cost equations, leading to an incorrect solution of \( x = \frac{1}{4} \) days. The correct rental duration for equal costs is **4 days**.