Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

Question

Two rental car companies are running specials this month. At Mateo's Rentals, customers will pay $51 to rent a mid-sized car for the first day, plus $3 for each additional day. At Campbell Rent-a-Car, the price for a mid-sized car is $46 for the first day and $8 for every additional day beyond that. At some point, renting from either one of the companies would cost a customer the same amount. How much would the customer pay? How many additional days would that take?

The customer would pay $
either way for
additional days.

Answer 1

To set up the problem, we need to create equations for the total cost from each rental car company based on the number of additional days rented.

Let's denote the number of additional days rented (beyond the first day) as \(d\).

For Mateo's Rentals, the cost \(C_M\) can be represented as: \[ C_M = 51 + 3d \]

For Campbell Rent-a-Car, the cost \(C_C\) can be represented as: \[ C_C = 46 + 8d \]

To find when the costs are equal, we set the equations equal to each other: \[ 51 + 3d = 46 + 8d \]

Now, we will solve for \(d\).

1. First, subtract \(3d\) from both sides: \[ 51 = 46 + 5d \]

2. Next, subtract \(46\) from both sides: \[ 5 = 5d \]

3. Now, divide both sides by \(5\): \[ d = 1 \]

So, the customer would need to rent an additional 1 day for the costs to be the same.

Now we substitute \(d\) back into either cost equation to find out how much the customer would pay for that time. Let's use Mateo's Rentals: \[ C_M = 51 + 3(1) = 51 + 3 = 54 \] We can also check with Campbell Rent-a-Car: \[ C_C = 46 + 8(1) = 46 + 8 = 54 \]

Both companies yield the same total cost.

Final answers:

The customer would pay $54 either way for 1 additional day.