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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ A rental car company is running two specials. Customers can pay $\$47$ to rent a compact car for the first day plus $\$5$ for each additional day, or they can rent the same car for $\$48$ the first day and $\$4$ for every additional day beyond that. Christina 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 Christina want? How much would Christina pay in total? $\newline$ For $\_\_\_\_$ additional days, Christina would pay $\$\_\_\_\_$ either way.

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Solve a system of equations using substitution: word problems

Full solution

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

\newline

A rental car company is running two specials. Customers can pay

\$47

to rent a compact car for the first day plus

\$5

for each additional day, or they can rent the same car for

\$48

the first day and

\$4

for every additional day beyond that. Christina 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 Christina want? How much would Christina pay in total?

\newline

For

\_\_\_\_

additional days, Christina would pay

\$\_\_\_\_

either way.

Define Equations: Step $1$ : Define the equations for the two rental specials. $\newline$ For the first special, the total cost $C_1$ is \(47\) for the first day plus $5$ for each additional day $x$ : $\newline$ $C_1 = 47 + 5x$ $\newline$ For the second special, the total cost $C_2$ is \(48\) for the first day plus $4$ for each additional day $x$ : $\newline$ $C_2 = 48 + 4x$
Set Equations Equal: Step $2$ : Set the equations equal to find when the costs are the same. $\newline$ $47 + 5x = 48 + 4x$ $\newline$ Subtract $47$ from both sides: $\newline$ $5x = 1 + 4x$ $\newline$ Subtract $4$ x from both sides: $\newline$ $x = 1$
Calculate Total Cost: Step $3$ : Calculate the total cost for $1$ additional day using either equation. $\newline$ Using $C_1$ : $\newline$ $C_1 = 47 + 5(1)$ $\newline$ $C_1 = 47 + 5$ $\newline$ $C_1 = 52$

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