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

Question

Mr. Norman, the owner of two car dealerships in Millersburg, is holding a contest to see which one can sell the most cars. Millersburg Cars has already sold 10 cars, and Norman's Autos has sold 7 cars. Going forward, the salespeople at Millersburg Cars think they can sell 2 cars per day, whereas the salespeople at Norman's Autos are aiming for sales of 5 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's define the variables:

• Let \( x \) be the number of days from now.
• Millersburg Cars has already sold 10 cars and sells 2 cars per day, so the total number of cars sold by Millersburg Cars after \( x \) days will be \( 10 + 2x \).
• Norman's Autos has already sold 7 cars and sells 5 cars per day, so the total number of cars sold by Norman's Autos after \( x \) days will be \( 7 + 5x \).

We can set up the following equation to find when both dealerships have sold the same number of cars:

\[ 10 + 2x = 7 + 5x \]

Now, let's solve for \( x \).

1. Start by subtracting \( 2x \) from both sides:

\[ 10 = 7 + 5x - 2x \] \[ 10 = 7 + 3x \]

2. Next, subtract 7 from both sides:

\[ 10 - 7 = 3x \] \[ 3 = 3x \]

3. Divide both sides by 3:

\[ x = 1 \]

Now we know it will take 1 day for the two dealerships to be tied.

Next, we can find out how many cars each lot will have sold after 1 day:

• For Millersburg Cars:

\[ 10 + 2(1) = 10 + 2 = 12 \]

• For Norman's Autos:

\[ 7 + 5(1) = 7 + 5 = 12 \]

After 1 day, both dealerships will have sold 12 cars.

So, to fill in the blanks:

In 1 day, the dealerships will each have sold 12 cars.