Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Two rental car companies are running specials this month. At Manuel's Rentals, customers will pay $\$45$ to rent a mid-sized car for the first day, plus $\$13$ for each additional day. At Wildgrove Rent-a-Car, the price for a mid-sized car is $\$46$ for the first day and $\$12$ 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? $\newline$ The customer would pay $\$$ _ either way for _ additional days.

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Grade 8

Solve a system of equations using substitution: word problems

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

\newline

Two rental car companies are running specials this month. At Manuel's Rentals, customers will pay

\$45

to rent a mid-sized car for the first day, plus

\$13

for each additional day. At Wildgrove Rent-a-Car, the price for a mid-sized car is

\$46

for the first day and

\$12

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?

\newline

The customer would pay

\$

_____ either way for _____ additional days.

Define Variables: Let's define the variables: $\newline$ Let $x$ be the number of additional days beyond the first day. $\newline$ Let $y$ be the total cost for renting the car. $\newline$ For Manuel's Rentals, the cost equation is: $\newline$ $y = 45 + 13x$ (since the first day costs $\$45$ and each additional day costs $\$13$ )
Cost Equations: For Wildgrove Rent-a-Car, the cost equation is: $\newline$ $y = 46 + 12x$ (since the first day costs $\$46$ and each additional day costs $\$12$ )
Solve System of Equations: Now we have a system of equations: $\newline$ $1$ ) $y = 45 + 13x$ $\newline$ $2$ ) $y = 46 + 12x$ $\newline$ We will solve this system using substitution. Since both equations equal $y$ , we can set them equal to each other: $\newline$ $45 + 13x = 46 + 12x$
Solve for x: Next, we solve for x: $\newline$ Subtract $12x$ from both sides: $\newline$ $45 + 13x - 12x = 46 + 12x - 12x$ $\newline$ $45 + x = 46$
Substitute $x$ : Subtract $45$ from both sides: $\newline$ $x = 46 - 45$ $\newline$ $x = 1$ $\newline$ So, the number of additional days is $1$ .
Find Total Cost: Now we substitute $x$ back into one of the original equations to find $y$ . We'll use the first equation: $\newline$ $y = 45 + 13x$ $\newline$ $y = 45 + 13(1)$ $\newline$ $y = 45 + 13$ $\newline$ $y = 58$ $\newline$ So, the total cost would be $\$58$ .