Question

Use your graphing calculator: Desmos Graphing CalculatorA company owns two dealerships, both of which sell cars and trucks. Dealership A sells a total of 225 cars and trucks. Dealership B sells twice as many cars and half as many trucks as the Dealership A, and sells a total of 300 cars and trucks. (4 points)a) An equation for the total cars and trucks for Dealership A: b) An equation for the total cars and trucks for Dealership B: c) How many cars did Dealership A sell? d) Given the fact that Dealership B sold half as many trucks as Dealership A, how many trucks did Dealership B sell?

Answer 1

Let's define the variables for the number of cars and trucks sold at Dealership A and Dealership B:

- Let \( c_A \) = number of cars sold at Dealership A
- Let \( t_A \) = number of trucks sold at Dealership A
- Let \( c_B \) = number of cars sold at Dealership B
- Let \( t_B \) = number of trucks sold at Dealership B

Now, we can break down the problem step by step:

### a) Equation for the total cars and trucks for Dealership A:
Given that Dealership A sells a total of 225 cars and trucks, we write the equation as:
\[
c_A + t_A = 225
\]

### b) Equation for the total cars and trucks for Dealership B:
According to the problem, Dealership B sells twice as many cars and half as many trucks as Dealership A. Based on this, the equations for cars and trucks sold at Dealership B can be expressed as:
\[
c_B = 2c_A
\]
\[
t_B = \frac{1}{2}t_A
\]
Additionally, we know that Dealership B sells a total of 300 cars and trucks:
\[
c_B + t_B = 300
\]

### Replace \( c_B \) and \( t_B \) in the total equation for Dealership B:
Substituting \( c_B \) and \( t_B \) into the equation gives us:
\[
2c_A + \frac{1}{2}t_A = 300
\]

### Now we have the following system of equations:
1. \( c_A + t_A = 225 \)
2. \( 2c_A + \frac{1}{2}t_A = 300 \)

### c) How many cars did Dealership A sell?
We can solve the system of equations to find the number of cars sold at Dealership A.

From the first equation, we can express \( t_A \):
\[
t_A = 225 - c_A
\]

Substituting this expression for \( t_A \) in the second equation:
\[
2c_A + \frac{1}{2}(225 - c_A) = 300
\]
Simplifying this:
\[
2c_A + 112.5 - \frac{1}{2}c_A = 300
\]
Combining terms:
\[
\frac{3}{2}c_A + 112.5 = 300
\]
Subtracting 112.5 from both sides:
\[
\frac{3}{2}c_A = 187.5
\]
Multiplying both sides by \( \frac{2}{3} \):
\[
c_A = 125
\]

### Answer for c): Dealership A sold 125 cars.

### d) Given that Dealership B sold half as many trucks as Dealership A, how many trucks did Dealership B sell?
First, we can find \( t_A \):
\[
t_A = 225 - c_A = 225 - 125 = 100
\]

Since Dealership B sells half as many trucks as Dealership A, we have:
\[
t_B = \frac{1}{2}t_A = \frac{1}{2}(100) = 50
\]

### Answer for d): Dealership B sold 50 trucks.

### Summary:
- a) \( c_A + t_A = 225 \)
- b) \( 2c_A + \frac{1}{2}t_A = 300 \)
- c) 125 cars sold by Dealership A.
- d) 50 trucks sold by Dealership B.