Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Mr. Livingston, the owner of two car dealerships in Centerville, is holding a contest to see which one can sell the most cars. Centerville Cars has already sold $17$ cars, and Livingston's Autos has sold $7$ cars. Going forward, the salespeople at Centerville Cars think they can sell $7$ cars per day, whereas the salespeople at Livingston's Autos are aiming for sales of $12$ cars per day. If the salespeople's predictions are accurate, it won't be long before the two dealerships are tied. How long will that take? How many cars will each lot have sold? $\newline$ In $\_$ days, the dealerships will each have sold $\_$ cars.

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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

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

17

cars, and Livingston's Autos has sold

7

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

7

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

12

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

\newline

\_

days, the dealerships will each have sold

\_

cars.

Define Variables: Let $x$ represent the number of days, and $y$ represent the number of cars sold. For Centerville Cars: Initial cars sold $= 17$ , Rate of selling $= 7$ cars/day. Equation: $y = 7x + 17$
Centerville Cars: For Livingston's Autos: Initial cars sold = $7$ , Rate of selling = $12$ cars/day. Equation: $y = 12x + 7$
Livingston's Autos: Set the equations equal to find when the dealerships will have sold the same number of cars: $7x + 17 = 12x + 7$
Set Equations Equal: Solve for $x$ : Subtract $7x$ from both sides: $17 = 5x + 7$
Solve for x: Subtract $7$ from both sides: $10 = 5x$
Substitute $x$ : Divide both sides by $5$ : $x = 2$
Final Result: Substitute $x = 2$ back into one of the original equations to find $y$ : Using $y = 7x + 17$ : $y = 7(2) + 17 = 14 + 17 = 31$