Question

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

Mr. Floyd, the owner of two car dealerships in Manchester, is holding a contest to see which one can sell the most cars. Manchester Cars has already sold 9 cars, and Floyd's Autos has sold 8 cars. Going forward, the salespeople at Manchester Cars think they can sell 9 cars per day, whereas the salespeople at Floyd's Autos are aiming for sales of 10 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?


In
days, the dealerships will each have sold
cars.

Answer 1

Let \( x \) represent the number of days after today. We can set up a system of equations to describe the total number of cars sold by each dealership over the next \( x \) days.

1. **Manchester Cars’ total sales** after \( x \) days:
\[
\text{Total sales of Manchester Cars} = 9 + 9x
\]

2. **Floyd's Autos' total sales** after \( x \) days:
\[
\text{Total sales of Floyd's Autos} = 8 + 10x
\]

To find the number of days \( x \) when both dealerships will have sold the same number of cars, we set the two total sales equal to each other:
\[
9 + 9x = 8 + 10x
\]

Now, we can solve this equation:

1. Subtract \( 9x \) from both sides:
\[
9 = 8 + x
\]

2. Subtract 8 from both sides:
\[
1 = x
\]

So, \( x = 1 \) day.

Next, we can calculate how many cars each dealership will have sold after 1 day:

- For Manchester Cars:
\[
9 + 9(1) = 9 + 9 = 18 \text{ cars}
\]

- For Floyd's Autos:
\[
8 + 10(1) = 8 + 10 = 18 \text{ cars}
\]

Therefore, both dealerships will have sold 18 cars.

In **1 day**, the dealerships will each have sold **18** cars.