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

Question

A rental car company is running two specials. Customers can pay $33 to rent a compact car for the first day plus $10 for each additional day, or they can rent the same car for $36 the first day and $9 for every additional day beyond that. Kari notices that, given the number of additional days she wants to rent the car for, the two specials are equivalent. How many additional days does Kari want? How much would Kari pay in total?

Answer 1

Let's define the variables:

• Let \( d \) be the number of additional days Kari wants to rent the car.

Now we can set up the equations for the total cost of renting the car under the two specials:

1. For the first special:

   • The cost for the first day is $33.
   • The cost for each additional day is $10.
   • Therefore, the total cost \( C_1 \) after \( d \) additional days is: \[ C_1 = 33 + 10d \]

2. For the second special:

   • The cost for the first day is $36.
   • The cost for each additional day is $9.
   • Therefore, the total cost \( C_2 \) after \( d \) additional days is: \[ C_2 = 36 + 9d \]

Since we want to find out when these two costs are equivalent, we set them equal to each other: \[ 33 + 10d = 36 + 9d \]

Now we will solve for \( d \):

1. Subtract \( 9d \) from both sides: \[ 33 + 10d - 9d = 36 \] \[ 33 + d = 36 \]
2. Subtract 33 from both sides: \[ d = 36 - 33 \] \[ d = 3 \]

Now that we have \( d = 3 \), we can find the total cost using either of the two equations. Let's use the first special's equation: \[ C_1 = 33 + 10d = 33 + 10(3) = 33 + 30 = 63 \]

Now we can fill in the blanks:

• Kari wants to rent the car for 3 additional days.
• Kari will pay $63 in total.