Question
Write and solve a system of equations. Only write the number, not the word days
at zippy rent a car, you rent a car for $25 per day, which a one time fee of $ 100. at speedy rent a car, you can rent a car for $30 per hour, with a one-time fee of $30 . write and solve a system of equations for which both car rental locations will cost the same.
at zippy rent a car, you rent a car for $25 per day, which a one time fee of $ 100. at speedy rent a car, you can rent a car for $30 per hour, with a one-time fee of $30 . write and solve a system of equations for which both car rental locations will cost the same.
Answers
GPT-4o mini
Let \( x \) represent the number of days you rent the car from Zippy Rent-A-Car, and let \( y \) represent the number of hours you rent the car from Speedy Rent-A-Car.
The cost for renting from Zippy Rent-A-Car can be expressed as:
\[
C_Z = 25x + 100
\]
For Speedy Rent-A-Car, the cost can be expressed as:
\[
C_S = 30y + 30
\]
To find when the costs are equal, we set the two equations equal to each other:
\[
25x + 100 = 30y + 30
\]
Now, we need to express one variable in terms of the other. Let's solve for \( y \):
\[
25x + 100 - 30 = 30y
\]
\[
25x + 70 = 30y
\]
\[
y = \frac{25x + 70}{30}
\]
This equation relates the number of days rented with the number of hours rented.
To solve the system, we can set values for \( x \) (number of days) and find corresponding \( y \) (number of hours). However, since one represents days and the other hours, we should assume a specific value for either \( x \) or \( y \).
Let's determine when it might make sense to rent for a certain number of days. Assume we want to rent for 2 days:
\[
x = 2
\]
Substituting into the equation:
\[
y = \frac{25(2) + 70}{30} = \frac{50 + 70}{30} = \frac{120}{30} = 4
\]
Thus, renting a car for 2 days at Zippy Rent-A-Car costs the same as renting for 4 hours at Speedy Rent-A-Car.
To check:
Cost at Zippy:
\[
C_Z = 25(2) + 100 = 50 + 100 = 150
\]
Cost at Speedy:
\[
C_S = 30(4) + 30 = 120 + 30 = 150
\]
Both costs are equal at $150.
Therefore, the system of equations has been solved with \( x = 2 \) and \( y = 4 \), meaning renting a car for 2 days at Zippy Rent-A-Car costs the same as renting for 4 hours at Speedy Rent-A-Car.
The cost for renting from Zippy Rent-A-Car can be expressed as:
\[
C_Z = 25x + 100
\]
For Speedy Rent-A-Car, the cost can be expressed as:
\[
C_S = 30y + 30
\]
To find when the costs are equal, we set the two equations equal to each other:
\[
25x + 100 = 30y + 30
\]
Now, we need to express one variable in terms of the other. Let's solve for \( y \):
\[
25x + 100 - 30 = 30y
\]
\[
25x + 70 = 30y
\]
\[
y = \frac{25x + 70}{30}
\]
This equation relates the number of days rented with the number of hours rented.
To solve the system, we can set values for \( x \) (number of days) and find corresponding \( y \) (number of hours). However, since one represents days and the other hours, we should assume a specific value for either \( x \) or \( y \).
Let's determine when it might make sense to rent for a certain number of days. Assume we want to rent for 2 days:
\[
x = 2
\]
Substituting into the equation:
\[
y = \frac{25(2) + 70}{30} = \frac{50 + 70}{30} = \frac{120}{30} = 4
\]
Thus, renting a car for 2 days at Zippy Rent-A-Car costs the same as renting for 4 hours at Speedy Rent-A-Car.
To check:
Cost at Zippy:
\[
C_Z = 25(2) + 100 = 50 + 100 = 150
\]
Cost at Speedy:
\[
C_S = 30(4) + 30 = 120 + 30 = 150
\]
Both costs are equal at $150.
Therefore, the system of equations has been solved with \( x = 2 \) and \( y = 4 \), meaning renting a car for 2 days at Zippy Rent-A-Car costs the same as renting for 4 hours at Speedy Rent-A-Car.