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

Mr. Hensley, the owner of two car dealerships in Oak Grove, is holding a contest to see which one can sell the most cars. Oak Grove Cars has already sold 8 cars, and Hensley's Autos has sold 6 cars. Going forward, the salespeople at Oak Grove Cars think they can sell 4 cars per day, whereas the salespeople at Hensley'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 many cars will each lot have sold? How long will that take?

Let's define the variables first:

- Let \( x \) represent the number of days it will take for the two dealerships to have sold the same number of cars.
- Let \( y \) represent the total number of cars sold by each dealership at that time.

Now, we can write the system of equations based on the information provided:

1. For Oak Grove Cars:
   \[
   y = 8 + 4x
   \]
   (They've already sold 8 cars and will sell 4 more cars per day.)

2. For Hensley's Autos:
   \[
   y = 6 + 5x
   \]
   (They've already sold 6 cars and will sell 5 more cars per day.)

To solve this system using substitution, we can set the two expressions for \( y \) equal to each other since both represent the total number of cars sold when the dealerships are tied.

\[
8 + 4x = 6 + 5x
\]

Now, solve for \( x \):

\[
8 - 6 = 5x - 4x
\]
\[
2 = x
\]

So, it will take \( x = 2 \) days for the two dealerships to have sold the same number of cars.

Now substitute \( x = 2 \) back into either equation to find \( y \). Let's use the equation for Oak Grove Cars:

\[
y = 8 + 4(2) = 8 + 8 = 16
\]

So, \( y = 16 \).

**Final Answers:**
- It will take **2 days** for the two dealerships to have sold the same number of cars.
- At that time, each lot will have sold **16 cars**.

Would you like more details or have any questions?

**Related Questions:**

1. What would happen if Oak Grove Cars could only sell 3 cars per day?
2. If Hensley's Autos increased their daily sales to 6 cars per day, how would that affect the outcome?
3. How many cars will each dealership have sold after 5 days if their sales rates remain unchanged?
4. If the initial sales were reversed (Oak Grove Cars sold 6 cars initially, and Hensley's Autos sold 8), how would the time to tie change?
5. What if a third dealership enters the contest, selling 4.5 cars per day and starting with 5 cars sold, how long until they tie with the others?

**Tip:** When dealing with systems of linear equations, substitution is a powerful method when one of the equations is easily solved for one variable.

Learn how to solve a system of linear equations involving a car sales contest between two dealerships. This problem involves using the substitution method to find out how long it will take for the dealerships to sell the same number of cars and the total sales at that time. Suitable for students in Grades 8-10.

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