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

Answer 1

The solution provided by the customer contains inaccuracies in the setup of the equations. Let’s go through the problem step-by-step to clarify and find the correct number of days for which the rental costs will be the same for both companies.

### Step 1: Write the Cost Equations

1. **Zippy Rent-a-Car:**
\[
C = 30 + 50x
\]
where \(C\) is the total cost and \(x\) is the number of days.

2. **Speedy Rent-a-Car:**
\[
C = 70 + 40x
\]

### Step 2: Set the Equations Equal to Each Other

To find when both costs are equal, we set the two equations equal to each other:
\[
30 + 50x = 70 + 40x
\]

### Step 3: Solve for \(x\)

1. Subtract \(40x\) from both sides:
\[
30 + 10x = 70
\]

2. Subtract 30 from both sides:
\[
10x = 40
\]

3. Divide by 10:
\[
x = 4
\]

### Conclusion

The rental costs for both Zippy Rent-a-Car and Speedy Rent-a-Car will be the same after **4 days**, not 1/4 day.

### Verification

Let’s verify by calculating the total costs for both rental companies after 4 days.

1. **Zippy Rent-a-Car:**
\[
C = 30 + 50(4) = 30 + 200 = 230
\]

2. **Speedy Rent-a-Car:**
\[
C = 70 + 40(4) = 70 + 160 = 230
\]

Both companies charge $230 after 4 days, confirming our solution is correct.