Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ Mr. Hardin, the owner of two car dealerships in Lanberry, is holding a contest to see which one can sell the most cars. Lanberry Cars has already sold $17$ cars, and Hardin's Autos has sold $11$ cars. Going forward, the salespeople at Lanberry Cars think they can sell $7$ cars per day, whereas the salespeople at Hardin's Autos are aiming for sales of $13$ cars per day. If the salespeople's predictions are accurate, it won't be long before the two dealerships are tied. How many cars will each lot have sold? How long will that take? $\newline$ The dealerships will each have sold _ cars in _ days.

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Solve a system of equations using any method: word problems

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

\newline

Mr. Hardin, the owner of two car dealerships in Lanberry, is holding a contest to see which one can sell the most cars. Lanberry Cars has already sold

17

cars, and Hardin's Autos has sold

11

cars. Going forward, the salespeople at Lanberry Cars think they can sell

7

cars per day, whereas the salespeople at Hardin's Autos are aiming for sales of

13

cars per day. If the salespeople's predictions are accurate, it won't be long before the two dealerships are tied. How many cars will each lot have sold? How long will that take?

\newline

The dealerships will each have sold _____ cars in _____ days.

Define Variables: Let's define the variables. Let $x$ be the number of days after which the two dealerships will have sold the same number of cars. Lanberry Cars has already sold $17$ cars and will sell $7$ more cars per day. Hardin's Autos has already sold $11$ cars and will sell $13$ more cars per day. $\newline$ First equation based on Lanberry Cars: $17 + 7x$ (total cars sold by Lanberry Cars after $x$ days) $\newline$ Second equation based on Hardin's Autos: $11 + 13x$ (total cars sold by Hardin's Autos after $x$ days)
Set Up Equations: Set up the system of equations to find when the two dealerships will have sold the same number of cars. $\newline$ Since both dealerships will have sold the same number of cars, we can set the expressions equal to each other: $\newline$ $17 + 7x = 11 + 13x$
Solve Equation: Solve the equation for $x$ . $17 + 7x = 11 + 13x$ Subtract $7x$ from both sides: $17 = 11 + 6x$ Subtract $11$ from both sides: $6 = 6x$ Divide both sides by $6$ : $x = 1$
Calculate Total Cars Sold: Calculate the total number of cars sold by each dealership after $x$ days. $\newline$ For Lanberry Cars: $17 + 7(1) = 17 + 7 = 24$ cars $\newline$ For Hardin's Autos: $11 + 13(1) = 11 + 13 = 24$ cars